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In July I will be spending a week at Park City, being one of the mini-course lecturers in the Graduate Summer School component of the Park City Summer Session on random matrices. I have chosen to give some lectures on least singular values of random matrices, the circular law, and the Lindeberg exchange method in random matrix theory; this is a slightly different set of topics than I had initially advertised (which was instead about the Lindeberg exchange method and the local relaxation flow method), but after consulting with the other mini-course lecturers I felt that this would be a more complementary set of topics. I have uploaded an draft of my lecture notes (some portion of which is derived from my monograph on the subject); as always, comments and corrections are welcome.
<I>[Update, June 23: notes revised and reformatted to PCMI format. -T.]</I>
Let be the divisor function. A classical application of the Dirichlet hyperbola method gives the asymptotic
where denotes the estimate as . Much better error estimates are possible here, but we will not focus on the lower order terms in this discussion. For somewhat idiosyncratic reasons I will interpret this estimate (and the other analytic number theory estimates discussed here) through the probabilistic lens. Namely, if is a random number selected uniformly between and , then the above estimate can be written as
that is to say the random variable has mean approximately . (But, somewhat paradoxically, this is not the median or mode behaviour of this random variable, which instead concentrates near , basically thanks to the Hardy-Ramanujan theorem.)
Now we turn to the pair correlations for a fixed positive integer . There is a classical computation of Ingham that shows that
The error term in (2) has been refined by many subsequent authors, as has the uniformity of the estimates in the aspect, as these topics are related to other questions in analytic number theory, such as fourth moment estimates for the Riemann zeta function; but we will not consider these more subtle features of the estimate here. However, we will look at the next term in the asymptotic expansion for (2) below the fold.
Using our probabilistic lens, the estimate (2) can be written as
From (1) (and the asymptotic negligibility of the shift by ) we see that the random variables and both have a mean of , so the additional factor of represents some arithmetic coupling between the two random variables.
Ingham's formula can be established in a number of ways. Firstly, one can expand out and use the hyperbola method (splitting into the cases and and removing the overlap). If one does so, one soon arrives at the task of having to estimate sums of the form
for various . For much less than this can be achieved using a further application of the hyperbola method, but for comparable to things get a bit more complicated, necessitating the use of non-trivial estimates on Kloosterman sums in order to obtain satisfactory control on error terms. A more modern approach proceeds using automorphic form methods, as discussed in this previous post. A third approach, which unfortunately is only heuristic at the current level of technology, is to apply the Hardy-Littlewood circle method (discussed in this previous post) to express (2) in terms of exponential sums for various frequencies . The contribution of "major arc" can be computed after a moderately lengthy calculation which yields the right-hand side of (2) (as well as the correct lower order terms that are currently being suppressed), but there does not appear to be an easy way to show directly that the "minor arc" contributions are of lower order, although the methods discussed previously do indirectly show that this is ultimately the case.
Each of the methods outlined above requires a fair amount of calculation, and it is not obvious while performing them that the factor will emerge at the end. One can at least explain the as a normalisation constant needed to balance the factor (at a heuristic level, at least). To see this through our probabilistic lens, introduce an independent copy of , then
using symmetry to order (discarding the diagonal case ) and making the change of variables , we see that (4) is heuristically consistent with (3) as long as the asymptotic mean of in is equal to . (This argument is not rigorous because there was an implicit interchange of limits present, but still gives a good heuristic "sanity check" of Ingham's formula.) Indeed, if denotes the asymptotic mean in , then we have (heuristically at least)
and we obtain the desired consistency after multiplying by .
This still however does not explain the presence of the factor. Intuitively it is reasonable that if has many prime factors, and has a lot of factors, then will have slightly more factors than average, because any common factor to and will automatically be acquired by . But how to quantify this effect?
One heuristic way to proceed is through analysis of local factors. Observe from the fundamental theorem of arithmetic that we can factor
where the product is over all primes , and is the local version of at (which in this case, is just one plus the –valuation of : ). Note that all but finitely many of the terms in this product will equal , so the infinite product is well-defined. In a similar fashion, we can factor
(or in terms of valuations, ). Heuristically, the Chinese remainder theorem suggests that the various factors behave like independent random variables, and so the correlation between and should approximately decouple into the product of correlations between the local factors and . And indeed we do have the following local version of Ingham's asymptotics:
Proposition 1 (Local Ingham asymptotics) For fixed and integer , we have
From the Euler formula
we see that
and so one can "explain" the arithmetic factor in Ingham's asymptotic as the product of the arithmetic factors in the (much easier) local Ingham asymptotics. Unfortunately we have the usual "local-global" problem in that we do not know how to rigorously derive the global asymptotic from the local ones; this problem is essentially the same issue as the problem of controlling the minor arc contributions in the circle method, but phrased in "physical space" language rather than "frequency space".
Remark 2 The relation between the local means and the global mean can also be seen heuristically through the application
of Mertens' theorem, where is Pólya's magic exponent, which serves as a useful heuristic limiting threshold in situations where the product of local factors is divergent.
Let us now prove this proposition. One could brute-force the computations by observing that for any fixed , the valuation is equal to with probability , and with a little more effort one can also compute the joint distribution of and , at which point the proposition reduces to the calculation of various variants of the geometric series. I however find it cleaner to proceed in a more recursive fashion (similar to how one can prove the geometric series formula by induction); this will also make visible the vague intuition mentioned previously about how common factors of and force to have a factor also.
It is first convenient to get rid of error terms by observing that in the limit , the random variable converges vaguely to a uniform random variable on the profinite integers , or more precisely that the pair converges vaguely to . Because of this (and because of the easily verified uniform integrability properties of and their powers), it suffices to establish the exact formulae
in the profinite setting (this setting will make it easier to set up the recursion).
We begin with (5). Observe that is coprime to with probability , in which case is equal to . Conditioning to the complementary probability event that is divisible by , we can factor where is also uniformly distributed over the profinite integers, in which event we have . We arrive at the identity
As and have the same distribution, the quantities and are equal, and (5) follows by a brief amount of high-school algebra.
We use a similar method to treat (6). First treat the case when is coprime to . Then we see that with probability , and are simultaneously coprime to , in which case . Furthermore, with probability , is divisible by and is not; in which case we can write as before, with and . Finally, in the remaining event with probability , is divisible by and is not; we can then write , so that and . Putting all this together, we obtain
and the claim (6) in this case follows from (5) and a brief computation (noting that in this case).
Now suppose that is divisible by , thus for some integer . Then with probability , and are simultaneously coprime to , in which case . In the remaining event, we can write , and then and . Putting all this together we have
which by (5) (and replacing by ) leads to the recursive relation
and (6) then follows by induction on the number of powers of .
The estimate (2) of Ingham was refined by Estermann, who obtained the more accurate expansion
for certain complicated but explicit coefficients . For instance, is given by the formula
where is the Euler-Mascheroni constant,
The formula for is similar but even more complicated. The error term was improved by Heath-Brown to ; it is conjectured (for instance by Conrey and Gonek) that one in fact has square root cancellation here, but this is well out of reach of current methods.
These lower order terms are traditionally computed either from a Dirichlet series approach (using Perron's formula) or a circle method approach. It turns out that a refinement of the above heuristics can also predict these lower order terms, thus keeping the calculation purely in physical space as opposed to the "multiplicative frequency space" of the Dirichlet series approach, or the "additive frequency space" of the circle method, although the computations are arguably as messy as the latter computations for the purposes of working out the lower order terms. We illustrate this just for the term below the fold.
Note: the following is a record of some whimsical mathematical thoughts and computations I had after doing some grading. It is likely that the sort of problems discussed here are in fact well studied in the appropriate literature; I would appreciate knowing of any links to such.
Suppose one assigns true-false questions on an examination, with the answers randomised so that each question is equally likely to have "true" as the correct answer as "false", with no correlation between different questions. Suppose that the students taking the examination must answer each question with exactly one of "true" or "false" (they are not allowed to skip any question). Then it is easy to see how to grade the exam: one can simply count how many questions each student answered correctly (i.e. each correct answer scores one point, and each incorrect answer scores zero points), and give that number as the final grade of the examination. More generally, one could assign some score of points to each correct answer and some score (possibly negative) of points to each incorrect answer, giving a total grade of points. As long as , this grade is simply an affine rescaling of the simple grading scheme and would serve just as well for the purpose of evaluating the students, as well as encouraging each student to answer the questions as correctly as possible.
In practice, though, a student will probably not know the answer to each individual question with absolute certainty. One can adopt a probabilistic model, where for a given student and a given question , the student may think that the answer to question is true with probability and false with probability , where is some quantity that can be viewed as a measure of confidence has in the answer (with being confident that the answer is true if is close to , and confident that the answer is false if is close to ); for simplicity let us assume that in 's probabilistic model, the answers to each question are independent random variables. Given this model, and assuming that the student wishes to maximise his or her expected grade on the exam, it is an easy matter to see that the optimal strategy for to take is to answer question true if and false if . (If , the student can answer arbitrarily.)
[Important note: here we are not using the term "confidence" in the technical sense used in statistics, but rather as an informal term for "subjective probability".]
This is fine as far as it goes, but for the purposes of evaluating how well the student actually knows the material, it provides only a limited amount of information, in particular we do not get to directly see the student's subjective probabilities for each question. If for instance answered out of questions correctly, was it because he or she actually knew the right answer for seven of the questions, or was it because he or she was making educated guesses for the ten questions that turned out to be slightly better than random chance? There seems to be no way to discern this if the only input the student is allowed to provide for each question is the single binary choice of true/false.
But what if the student were able to give probabilistic answers to any given question? That is to say, instead of being forced to answer just "true" or "false" for a given question , the student was allowed to give answers such as " confident that the answer is true" (and hence confidence the answer is false). Such answers would give more insight as to how well the student actually knew the material; in particular, we would theoretically be able to actually see the student's subjective probabilities .
But now it becomes less clear what the right grading scheme to pick is. Suppose for instance we wish to extend the simple grading scheme in which an correct answer given in confidence is awarded one point. How many points should one award a correct answer given in confidence? How about an incorrect answer given in confidence (or equivalently, a correct answer given in confidence)?
Mathematically, one could design a grading scheme by selecting some grading function and then awarding a student points whenever they indicate the correct answer with a confidence of . For instance, if the student was confident that the answer was "true" (and hence confident that the answer was "false"), then this grading scheme would award the student points if the correct answer actually was "true", and points if the correct answer actually was "false". One can then ask the question of what functions would be "best" for this scheme?
Intuitively, one would expect that should be monotone increasing – one should be rewarded more for being correct with high confidence, than correct with low confidence. On the other hand, some sort of "partial credit" should still be assigned in the latter case. One obvious proposal is to just use a linear grading function – thus for instance a correct answer given with confidence might be worth points. But is this the "best" option?
To make the problem more mathematically precise, one needs an objective criterion with which to evaluate a given grading scheme. One criterion that one could use here is the avoidance of perverse incentives. If a grading scheme is designed badly, a student may end up overstating or understating his or her confidence in an answer in order to optimise the (expected) grade: the optimal level of confidence for a student to report on a question may differ from that student's subjective confidence . So one could ask to design a scheme so that is always equal to , so that the incentive is for the student to honestly report his or her confidence level in the answer.
This turns out to give a precise constraint on the grading function . If a student thinks that the answer to a question is true with probability and false with probability , and enters in an answer of "true" with confidence (and thus "false" with confidence ), then student would expect a grade of
on average for this question. To maximise this expected grade (assuming differentiability of , which is a reasonable hypothesis for a partial credit grading scheme), one performs the usual maneuvre of differentiating in the independent variable and setting the result to zero, thus obtaining
In order to avoid perverse incentives, the maximum should occur at , thus we should have
for all . This suggests that the function should be constant. (Strictly speaking, it only gives the weaker constraint that is symmetric around ; but if one generalised the problem to allow for multiple-choice questions with more than two possible answers, with a grading scheme that depended only on the confidence assigned to the correct answer, the same analysis would in fact force to be constant in ; we leave this computation to the interested reader.) In other words, should be of the form for some ; by monotonicity we expect to be positive. If we make the normalisation (so that no points are awarded for a split in confidence between true and false) and , one arrives at the grading scheme
Thus, if a student believes that an answer is "true" with confidence and "false" with confidence , he or she will be awarded points when the correct answer is "true", and points if the correct answer is "false". The following table gives some illustrative values for this scheme:
|Confidence that answer is "true"
|Points awarded if answer is "true"
|Points awarded if answer is "false"
Note the large penalties for being extremely confident of an answer that ultimately turns out to be incorrect; in particular, answers of confidence should be avoided unless one really is absolutely certain as to the correctness of one's answer.
The total grade given under such a scheme to a student who answers each question to be "true" with confidence , and "false" with confidence , is
This grade can also be written as
is the likelihood of the student 's subjective probability model, given the outcome of the correct answers. Thus the grade system here has another natural interpretation, as being an affine rescaling of the log-likelihood. The incentive is thus for the student to maximise the likelihood of his or her own subjective model, which aligns well with standard practices in statistics. From the perspective of Bayesian probability, the grade given to a student can then be viewed as a measurement (in logarithmic scale) of how much the posterior probability that the student's model was correct has improved over the prior probability.
One could propose using the above grading scheme to evaluate predictions to binary events, such as an upcoming election with only two viable candidates, to see in hindsight just how effective each predictor was in calling these events. One difficulty in doing so is that many predictions do not come with explicit probabilities attached to them, and attaching a default confidence level of to any prediction made without any such qualification would result in an automatic grade of if even one of these predictions turned out to be incorrect. But perhaps if a predictor refuses to attach confidence level to his or her predictions, one can assign some default level of confidence to these predictions, and then (using some suitable set of predictions from this predictor as "training data") find the value of that maximises this predictor's grade. This level can then be used going forward as the default level of confidence to apply to any future predictions from this predictor.
The above grading scheme extends easily enough to multiple-choice questions. But one question I had trouble with was how to deal with uncertainty, in which the student does not know enough about a question to venture even a probability of being true or false. Here, it is natural to allow a student to leave a question blank (i.e. to answer "I don't know"); a more advanced option would be to allow the student to enter his or her confidence level as an interval range (e.g. "I am between and confident that the answer is "true""). But now I do not have a good proposal for a grading scheme; once there is uncertainty in the student's subjective model, the problem of that student maximising his or her expected grade becomes ill-posed due to the "unknown unknowns", and so the previous criterion of avoiding perverse incentives becomes far less useful.
When teaching mathematics, the traditional method of lecturing in front of a blackboard is still hard to improve upon, despite all the advances in modern technology. However, there are some nice things one can do in an electronic medium, such as this blog. Here, I would like to experiment with the ability to animate images, which I think can convey some mathematical concepts in ways that cannot be easily replicated by traditional static text and images. Given that many readers may find these animations annoying, I am placing the rest of the post below the fold.
In the previous set of notes we established the central limit theorem, which we formulate here as follows:
Theorem 1 (Central limit theorem) Let be iid copies of a real random variable of mean and variance , and write . Then, for any fixed , we have
This is however not the end of the matter; there are many variants, refinements, and generalisations of the central limit theorem, and the purpose of this set of notes is to present a small sample of these variants.
First of all, the above theorem does not quantify the rate of convergence in (1). We have already addressed this issue to some extent with the Berry-Esséen theorem, which roughly speaking gives a convergence rate of uniformly in if we assume that has finite third moment. However there are still some quantitative versions of (1) which are not addressed by the Berry-Esséen theorem. For instance one may be interested in bounding the large deviation probabilities
in the setting where grows with . Chebyshev's inequality gives an upper bound of for this quantity, but one can often do much better than this in practice. For instance, the central limit theorem (1) suggests that this probability should be bounded by something like ; however, this theorem only kicks in when is very large compared with . For instance, if one uses the Berry-Esséen theorem, one would need as large as or so to reach the desired bound of , even under the assumption of finite third moment. Basically, the issue is that convergence-in-distribution results, such as the central limit theorem, only really control the typical behaviour of statistics in ; they are much less effective at controlling the very rare outlier events in which the statistic strays far from its typical behaviour. Fortunately, there are large deviation inequalities (or concentration of measure inequalities) that do provide exponential type bounds for quantities such as (2), which are valid for both small and large values of . A basic example of this is the Chernoff bound that made an appearance in Exercise 47 of Notes 4; here we give some further basic inequalities of this type, including versions of the Bennett and Hoeffding inequalities.
In the other direction, we can also look at the fine scale behaviour of the sums by trying to control probabilities such as
where is now bounded (but can grow with ). The central limit theorem predicts that this quantity should be roughly , but even if one is able to invoke the Berry-Esséen theorem, one cannot quite see this main term because it is dominated by the error term in Berry-Esséen. There is good reason for this: if for instance takes integer values, then also takes integer values, and can vanish when is less than and is slightly larger than an integer. However, this turns out to essentially be the only obstruction; if does not lie in a lattice such as , then we can establish a local limit theorem controlling (3), and when does take values in a lattice like , there is a discrete local limit theorem that controls probabilities such as . Both of these limit theorems will be proven by the Fourier-analytic method used in the previous set of notes.
We also discuss other limit theorems in which the limiting distribution is something other than the normal distribution. Perhaps the most common example of these theorems is the Poisson limit theorems, in which one sums a large number of indicator variables (or approximate indicator variables), each of which is rarely non-zero, but which collectively add up to a random variable of medium-sized mean. In this case, it turns out that the limiting distribution should be a Poisson random variable; this again is an easy application of the Fourier method. Finally, we briefly discuss limit theorems for other stable laws than the normal distribution, which are suitable for summing random variables of infinite variance, such as the Cauchy distribution.
Finally, we mention a very important class of generalisations to the CLT (and to the variants of the CLT discussed in this post), in which the hypothesis of joint independence between the variables is relaxed, for instance one could assume only that the form a martingale. Many (though not all) of the proofs of the CLT extend to these more general settings, and this turns out to be important for many applications in which one does not expect joint independence. However, we will not discuss these generalisations in this course, as they are better suited for subsequent courses in this series when the theory of martingales, conditional expectation, and related tools are developed.
Let be iid copies of an absolutely integrable real scalar random variable , and form the partial sums . As we saw in the last set of notes, the law of large numbers ensures that the empirical averages converge (both in probability and almost surely) to a deterministic limit, namely the mean of the reference variable . Furthermore, under some additional moment hypotheses on the underlying variable , we can obtain square root cancellation for the fluctuation of the empirical average from the mean. To simplify the calculations, let us first restrict to the case of mean zero and variance one, thus
Then, as computed in previous notes, the normalised fluctuation also has mean zero and variance one:
This and Chebyshev's inequality already indicates that the "typical" size of is , thus for instance goes to zero in probability for any that goes to infinity as . If we also have a finite fourth moment , then the calculations of the previous notes also give a fourth moment estimate
From this and the Paley-Zygmund inequality (Exercise 42 of Notes 1) we also get some lower bound for of the form
for some absolute constant and for sufficiently large; this indicates in particular that does not converge in any reasonable sense to something finite for any that goes to infinity.
The question remains as to what happens to the ratio itself, without multiplying or dividing by any factor . A first guess would be that these ratios converge in probability or almost surely, but this is unfortunately not the case:
Proposition 1 Let be iid copies of an absolutely integrable real scalar random variable with mean zero, variance one, and finite fourth moment, and write . Then the random variables do not converge in probability or almost surely to any limit, and neither does any subsequence of these random variables.
Proof: Suppose for contradiction that some sequence converged in probability or almost surely to a limit . By passing to a further subsequence we may assume that the convergence is in the almost sure sense. Since all of the have mean zero, variance one, and bounded fourth moment, Theorem 24 of Notes 1 implies that the limit also has mean zero and variance one. On the other hand, is a tail random variable and is thus almost surely constant by the Kolmogorov zero-one law from Notes 3. Since constants have variance zero, we obtain the required contradiction.
Nevertheless there is an important limit for the ratio , which requires one to replace the notions of convergence in probability or almost sure convergence by the weaker concept of convergence in distribution.
Definition 2 (Vague convergence and convergence in distribution) Let be a locally compact Hausdorff topological space with the Borel -algebra. A sequence of finite measures on is said to converge vaguely to another finite measure if one has
as for all continuous compactly supported functions . (Vague convergence is also known as weak convergence, although strictly speaking the terminology weak-* convergence would be more accurate.) A sequence of random variables taking values in is said to converge in distribution (or converge weakly or converge in law) to another random variable if the distributions converge vaguely to the distribution , or equivalently if
as for all continuous compactly supported functions .
One could in principle try to extend this definition beyond the locally compact Hausdorff setting, but certain pathologies can occur when doing so (e.g. failure of the Riesz representation theorem), and we will never need to consider vague convergence in spaces that are not locally compact Hausdorff, so we restrict to this setting for simplicity.
Note that the notion of convergence in distribution depends only on the distribution of the random variables involved. One consequence of this is that convergence in distribution does not produce unique limits: if converges in distribution to , and has the same distribution as , then also converges in distribution to . However, limits are unique up to equivalence in distribution (this is a consequence of the Riesz representation theorem, discussed for instance in this blog post). As a consequence of the insensitivity of convergence in distribution to equivalence in distribution, we may also legitimately talk about convergence of distribution of a sequence of random variables to another random variable even when all the random variables and involved are being modeled by different probability spaces (e.g. each is modeled by , and is modeled by , with no coupling presumed between these spaces). This is in contrast to the stronger notions of convergence in probability or almost sure convergence, which require all the random variables to be modeled by a common probability space. Also, by an abuse of notation, we can say that a sequence of random variables converges in distribution to a probability measure , when converges vaguely to . Thus we can talk about a sequence of random variables converging in distribution to a uniform distribution, a gaussian distribution, etc..
From the dominated convergence theorem (available for both convergence in probability and almost sure convergence) we see that convergence in probability or almost sure convergence implies convergence in distribution. The converse is not true, due to the insensitivity of convergence in distribution to equivalence in distribution; for instance, if are iid copies of a non-deterministic scalar random variable , then the trivially converge in distribution to , but will not converge in probability or almost surely (as one can see from the zero-one law). However, there are some partial converses that relate convergence in distribution to convergence in probability; see Exercise 10 below.
Remark 3 The notion of convergence in distribution is somewhat similar to the notion of convergence in the sense of distributions that arises in distribution theory (discussed for instance in this previous blog post), however strictly speaking the two notions of convergence are distinct and should not be confused with each other, despite the very similar names.
The notion of convergence in distribution simplifies in the case of real scalar random variables:
Proposition 4 Let be a sequence of scalar random variables, and let be another scalar random variable. Then the following are equivalent:
- (i) converges in distribution to .
- (ii) converges to for each continuity point of (i.e. for all real numbers at which is continuous). Here is the cumulative distribution function of .
Proof: First suppose that converges in distribution to , and is continuous at . For any , one can find a such that
for every . One can also find an larger than such that and . Thus
Let be a continuous function supported on that equals on . Then by the above discussion we have
and hence
for large enough . In particular
A similar argument, replacing with a continuous function supported on that equals on gives
for large enough. Putting the two estimates together gives
for large enough; sending , we obtain the claim.
Conversely, suppose that converges to at every continuity point of . Let be a continuous compactly supported function, then it is uniformly continuous. As is monotone increasing, it can only have countably many points of discontinuity. From these two facts one can find, for any , a simple function for some that are points of continuity of , and real numbers , such that for all . Thus
Similarly for replaced by . Subtracting and taking limit superior, we conclude that
and on sending , we obtain that converges in distribution to as claimed.
The restriction to continuity points of is necessary. Consider for instance the deterministic random variables , then converges almost surely (and hence in distribution) to , but does not converge to .
Example 5 For any natural number , let be a discrete random variable drawn uniformly from the finite set , and let be the continuous random variable drawn uniformly from . Then converges in distribution to . Thus we see that a continuous random variable can emerge as the limit of discrete random variables.
Example 6 For any natural number , let be a continuous random variable drawn uniformly from , then converges in distribution to the deterministic real number . Thus we see that discrete (or even deterministic) random variables can emerge as the limit of continuous random variables.
Exercise 7 (Portmanteau theorem) Show that the properties (i) and (ii) in Proposition 4 are also equivalent to the following three statements:
- (iii) One has for all closed sets .
- (iv) One has for all open sets .
- (v) For any Borel set whose topological boundary is such that , one has .
(Note: to prove this theorem, you may wish to invoke Urysohn's lemma. To deduce (iii) from (i), you may wish to start with the case of compact .)
We can now state the famous central limit theorem:
Theorem 8 (Central limit theorem) Let be iid copies of a scalar random variable of finite mean and finite non-zero variance . Let . Then the random variables converges in distribution to a random variable with the standard normal distribution (that is to say, a random variable with probability density function ). Thus, by abuse of notation
In the normalised case when has mean zero and unit variance, this simplifies to
Using Proposition 4 (and the fact that the cumulative distribution function associated to is continuous, the central limit theorem is equivalent to asserting that
as for any , or equivalently that
Informally, one can think of the central limit theorem as asserting that approximately behaves like it has distribution for large , where is the normal distribution with mean and variance , that is to say the distribution with probability density function . The integrals can be written in terms of the error function as .
The central limit theorem is a basic example of the universality phenomenon in probability – many statistics involving a large system of many independent (or weakly dependent) variables (such as the normalised sums ) end up having a universal asymptotic limit (in this case, the normal distribution), regardless of the precise makeup of the underlying random variable that comprised that system. Indeed, the universality of the normal distribution is such that it arises in many other contexts than the fluctuation of iid random variables; the central limit theorem is merely the first place in probability theory where it makes a prominent appearance.
We will give several proofs of the central limit theorem in these notes; each of these proofs has their advantages and disadvantages, and can each extend to prove many further results beyond the central limit theorem. We first give Lindeberg's proof of the central limit theorem, based on exchanging (or swapping) each component of the sum in turn. This proof gives an accessible explanation as to why there should be a universal limit for the central limit theorem; one then computes directly with gaussians to verify that it is the normal distribution which is the universal limit. Our second proof is the most popular one taught in probability texts, namely the Fourier-analytic proof based around the concept of the characteristic function of a real random variable . Thanks to the powerful identities and other results of Fourier analysis, this gives a quite short and direct proof of the central limit theorem, although the arguments may seem rather magical to readers who are not already familiar with Fourier methods. Finally, we give a proof based on the moment method, in the spirit of the arguments in the previous notes; this argument is more combinatorial, but is straightforward and is particularly robust, in particular being well equipped to handle some dependencies between components; we will illustrate this by proving the Erdos-Kac law in number theory by this method. Some further discussion of the central limit theorem (including some further proofs, such as one based on Stein's method) can be found in this blog post. Some further variants of the central limit theorem, such as local limit theorems, stable laws, and large deviation inequalities, will be discussed in the next (and final) set of notes.
The following exercise illustrates the power of the central limit theorem, by establishing combinatorial estimates which would otherwise require the use of Stirling's formula to establish.
Exercise 9 (De Moivre-Laplace theorem) Let be a Bernoulli random variable, taking values in with , thus has mean and variance . Let be iid copies of , and write .
- (i) Show that takes values in with . (This is an example of a binomial distribution.)
- (ii) Assume Stirling's formula
where is a function of that goes to zero as . (A proof of this formula may be found in this previous blog post.) Using this formula, and without using the central limit theorem, show that
as for any fixed real numbers .
The above special case of the central limit theorem was first established by de Moivre and Laplace.
We close this section with some basic facts about convergence of distribution that will be useful in the sequel.
Exercise 10 Let , be sequences of real random variables, and let be further real random variables.
- (i) If is deterministic, show that converges in distribution to if and only if converges in probability to .
- (ii) Suppose that is independent of for each , and independent of . Show that converges in distribution to if and only if converges in distribution to and converges in distribution to . (The shortest way to prove this is by invoking the Stone-Weierstrass theorem, but one can also proceed by proving some version of Proposition 4.) What happens if the independence hypothesis is dropped?
- (iii) If converges in distribution to , show that for every there exists such that for all sufficiently large . (That is to say, is a tight sequence of random variables.)
- (iv) Show that converges in distribution to if and only if, after extending the probability space model if necessary, one can find copies and of and respectively such that converges almost surely to . (Hint: use the Skorohod representation, Exercise 29 of Notes 0.)
- (v) If converges in distribution to , and is continuous, show that converges in distribution to . Generalise this claim to the case when takes values in an arbitrary locally compact Hausdorff space.
- (vi) (Slutsky's theorem) If converges in distribution to , and converges in probability to a deterministic limit , show that converges in distribution to , and converges in distribution to . (Hint: either use (iv), or else use (iii) to control some error terms.) This statement combines particularly well with (i). What happens if is not assumed to be deterministic?
- (vii) (Fatou lemma) If is continuous, and converges in distribution to , show that .
- (viii) (Bounded convergence) If is continuous and bounded, and converges in distribution to , show that .
- (ix) (Dominated convergence) If converges in distribution to , and there is an absolutely integrable such that almost surely for all , show that .
For future reference we also mention (but will not prove) Prokhorov's theorem that gives a partial converse to part (iii) of the above exercise:
Theorem 11 (Prokhorov's theorem) Let be a sequence of real random variables which is tight (that is, for every there exists such that for all sufficiently large ). Then there exists a subsequence which converges in distribution to some random variable (which may possibly be modeled by a different probability space model than the .)
The proof of this theorem relies on the Riesz representation theorem, and is beyond the scope of this course; but see for instance Exercise 29 of this previous blog post. (See also the closely related Helly selection theorem, covered in Exercise 30 of the same post.)
One of the major activities in probability theory is studying the various statistics that can be produced from a complex system with many components. One of the simplest possible systems one can consider is a finite sequence or an infinite sequence of jointly independent scalar random variables, with the case when the are also identically distributed (i.e. the are iid) being a model case of particular interest. (In some cases one may consider a triangular array of scalar random variables, rather than a finite or infinite sequence.) There are many statistics of such sequences that one can study, but one of the most basic such statistics are the partial sums
The first fundamental result about these sums is the law of large numbers (or LLN for short), which comes in two formulations, weak (WLLN) and strong (SLLN). To state these laws, we first must define the notion of convergence in probability.
Definition 1 Let be a sequence of random variables taking values in a separable metric space (e.g. the could be scalar random variables, taking values in or ), and let be another random variable taking values in . We say that converges in probability to if, for every radius , one has as . Thus, if are scalar, we have converging to in probability if as for any given .
The measure-theoretic analogue of convergence in probability is convergence in measure.
It is instructive to compare the notion of convergence in probability with almost sure convergence. it is easy to see that converges almost surely to if and only if, for every radius , one has as ; thus, roughly speaking, convergence in probability is good for controlling how a single random variable is close to its putative limiting value , while almost sure convergence is good for controlling how the entire tail of a sequence of random variables is close to its putative limit .
We have the following easy relationships between convergence in probability and almost sure convergence:
Exercise 2 Let be a sequence of scalar random variables, and let be another scalar random variable.
- (i) If almost surely, show that in probability. Give a counterexample to show that the converse does not necessarily hold.
- (ii) Suppose that for all . Show that almost surely. Give a counterexample to show that the converse does not necessarily hold.
- (iii) If in probability, show that there is a subsequence of the such that almost surely.
- (iv) If are absolutely integrable and as , show that in probability. Give a counterexample to show that the converse does not necessarily hold.
- (v) (Urysohn subsequence principle) Suppose that every subsequence of has a further subsequence that converges to in probability. Show that also converges to in probability.
- (vi) Does the Urysohn subsequence principle still hold if "in probability" is replaced with "almost surely" throughout?
- (vii) If converges in probability to , and or is continuous, show that converges in probability to . More generally, if for each , is a sequence of scalar random variables that converge in probability to , and or is continuous, show that converges in probability to . (Thus, for instance, if and converge in probability to and respectively, then and converge in probability to and respectively.
- (viii) (Fatou's lemma for convergence in probability) If are non-negative and converge in probability to , show that .
- (ix) (Dominated convergence in probability) If converge in probability to , and one almost surely has for all and some absolutely integrable , show that converges to .
Exercise 3 Let be a sequence of scalar random variables converging in probability to another random variable .
- (i) Suppose that there is a random variable which is independent of for each individual . Show that is also independent of .
- (ii) Suppose that the are jointly independent. Show that is almost surely constant (i.e. there is a deterministic scalar such that almost surely).
We can now state the weak and strong law of large numbers, in the model case of iid random variables.
Theorem 4 (Law of large numbers, model case) Let be an iid sequence of copies of an absolutely integrable random variable (thus the are independent and all have the same distribution as ). Write , and for each natural number , let denote the random variable .
- (i) (Weak law of large numbers) The random variables converge in probability to .
- (ii) (Strong law of large numbers) The random variables converge almost surely to .
Informally: if are iid with mean , then for large. Clearly the strong law of large numbers implies the weak law, but the weak law is easier to prove (and has somewhat better quantitative estimates). There are several variants of the law of large numbers, for instance when one drops the hypothesis of identical distribution, or when the random variable is not absolutely integrable, or if one seeks more quantitative bounds on the rate of convergence; we will discuss some of these variants below the fold.
It is instructive to compare the law of large numbers with what one can obtain from the Kolmogorov zero-one law, discussed in Notes 2. Observe that if the are real-valued, then the limit superior and are tail random variables in the sense that they are not affected if one changes finitely many of the ; in particular, events such as are tail events for any . From this and the zero-one law we see that there must exist deterministic quantities such that and almost surely. The strong law of large numbers can then be viewed as the assertion that when is absolutely integrable. On the other hand, the zero-one law argument does not require absolute integrability (and one can replace the denominator by other functions of that go to infinity as ).
The law of large numbers asserts, roughly speaking, that the theoretical expectation of a random variable can be approximated by taking a large number of independent samples of and then forming the empirical mean . This ability to approximate the theoretical statistics of a probability distribution through empirical data is one of the basic starting points for mathematical statistics, though this is not the focus of the course here. The tendency of statistics such as to cluster closely around their mean value is the simplest instance of the concentration of measure phenomenon, which is of tremendous significance not only within probability, but also in applications of probability to disciplines such as statistics, theoretical computer science, combinatorics, random matrix theory and high dimensional geometry. We will not discuss these topics much in this course, but see this previous blog post for some further discussion.
There are several ways to prove the law of large numbers (in both forms). One basic strategy is to use the moment method – controlling statistics such as by computing moments such as the mean , variance , or higher moments such as for . The joint independence of the make such moments fairly easy to compute, requiring only some elementary combinatorics. A direct application of the moment method typically requires one to make a finite moment assumption such as , but as we shall see, one can reduce fairly easily to this case by a truncation argument.
For the strong law of large numbers, one can also use methods relating to the theory of martingales, such as stopping time arguments and maximal inequalities; we present some classical arguments of Kolmogorov in this regard.
In the previous set of notes, we constructed the measure-theoretic notion of the Lebesgue integral, and used this to set up the probabilistic notion of expectation on a rigorous footing. In this set of notes, we will similarly construct the measure-theoretic concept of a product measure (restricting to the case of probability measures to avoid unnecessary techncialities), and use this to set up the probabilistic notion of independence on a rigorous footing. (To quote Durrett: "measure theory ends and probability theory begins with the definition of independence.") We will be able to take virtually any collection of random variables (or probability distributions) and couple them together to be independent via the product measure construction, though for infinite products there is the slight technicality (a requirement of the Kolmogorov extension theorem) that the random variables need to range in standard Borel spaces. This is not the only way to couple together such random variables, but it is the simplest and the easiest to compute with in practice, as we shall see in the next few sets of notes.
In Notes 0, we introduced the notion of a measure space , which includes as a special case the notion of a probability space. By selecting one such probability space as a sample space, one obtains a model for random events and random variables, with random events being modeled by measurable sets in , and random variables taking values in a measurable space being modeled by measurable functions . We then defined some basic operations on these random events and variables:
- Given events , we defined the conjunction , the disjunction , and the complement . For countable families of events, we similarly defined and . We also defined the empty event and the sure event , and what it meant for two events to be equal.
- Given random variables in ranges respectively, and a measurable function , we defined the random variable in range . (As the special case of this, every deterministic element of was also a random variable taking values in .) Given a relation , we similarly defined the event . Conversely, given an event , we defined the indicator random variable . Finally, we defined what it meant for two random variables to be equal.
- Given an event , we defined its probability .
These operations obey various axioms; for instance, the boolean operations on events obey the axioms of a Boolean algebra, and the probabilility function obeys the Kolmogorov axioms. However, we will not focus on the axiomatic approach to probability theory here, instead basing the foundations of probability theory on the sample space models as discussed in Notes 0. (But see this previous post for a treatment of one such axiomatic approach.)
It turns out that almost all of the other operations on random events and variables we need can be constructed in terms of the above basic operations. In particular, this allows one to safely extend the sample space in probability theory whenever needed, provided one uses an extension that respects the above basic operations; this is an important operation when one needs to add new sources of randomness to an existing system of events and random variables, or to couple together two separate such systems into a joint system that extends both of the original systems. We gave a simple example of such an extension in the previous notes, but now we give a more formal definition:
Definition 1 Suppose that we are using a probability space as the model for a collection of events and random variables. An extension of this probability space is a probability space , together with a measurable map (sometimes called the factor map) which is probability-preserving in the sense that
for all . (Caution: this does not imply that for all – why not?)
An event which is modeled by a measurable subset in the sample space , will be modeled by the measurable set in the extended sample space . Similarly, a random variable taking values in some range that is modeled by a measurable function in , will be modeled instead by the measurable function in . We also allow the extension to model additional events and random variables that were not modeled by the original sample space (indeed, this is one of the main reasons why we perform extensions in probability in the first place).
Thus, for instance, the sample space in Example 3 of the previous post is an extension of the sample space in that example, with the factor map given by the first coordinate projection . One can verify that all of the basic operations on events and random variables listed above are unaffected by the above extension (with one caveat, see remark below). For instance, the conjunction of two events can be defined via the original model by the formula
or via the extension via the formula
The two definitions are consistent with each other, thanks to the obvious set-theoretic identity
Similarly, the assumption (1) is precisely what is needed to ensure that the probability of an event remains unchanged when one replaces a sample space model with an extension. We leave the verification of preservation of the other basic operations described above under extension as exercises to the reader.
Remark 2 There is one minor exception to this general rule if we do not impose the additional requirement that the factor map is surjective. Namely, for non-surjective , it can become possible that two events are unequal in the original sample space model, but become equal in the extension (and similarly for random variables), although the converse never happens (events that are equal in the original sample space always remain equal in the extension). For instance, let be the discrete probability space with and , and let be the discrete probability space with , and non-surjective factor map defined by . Then the event modeled by in is distinct from the empty event when viewed in , but becomes equal to that event when viewed in . Thus we see that extending the sample space by a non-surjective factor map can identify previously distinct events together (though of course, being probability preserving, this can only happen if those two events were already almost surely equal anyway). This turns out to be fairly harmless though; while it is nice to know if two given events are equal, or if they differ by a non-null event, it is almost never useful to know that two events are unequal if they are already almost surely equal. Alternatively, one can add the additional requirement of surjectivity in the definition of an extension, which is also a fairly harmless constraint to impose (this is what I chose to do in this previous set of notes).
Roughly speaking, one can define probability theory as the study of those properties of random events and random variables that are model-independent in the sense that they are preserved by extensions. For instance, the cardinality of the model of an event is not a concept within the scope of probability theory, as it is not preserved by extensions: continuing Example 3 from Notes 0, the event that a die roll is even is modeled by a set of cardinality in the original sample space model , but by a set of cardinality in the extension. Thus it does not make sense in the context of probability theory to refer to the "cardinality of an event ".
On the other hand, the supremum of a collection of random variables in the extended real line is a valid probabilistic concept. This can be seen by manually verifying that this operation is preserved under extension of the sample space, but one can also see this by defining the supremum in terms of existing basic operations. Indeed, note from Exercise 24 of Notes 0 that a random variable in the extended real line is completely specified by the threshold events for ; in particular, two such random variables are equal if and only if the events and are surely equal for all . From the identity
we thus see that one can completely specify in terms of using only the basic operations provided in the above list (and in particular using the countable conjunction .) Of course, the same considerations hold if one replaces supremum, by infimum, limit superior, limit inferior, or (if it exists) the limit.
In this set of notes, we will define some further important operations on scalar random variables, in particular the expectation of these variables. In the sample space models, expectation corresponds to the notion of integration on a measure space. As we will need to use both expectation and integration in this course, we will thus begin by quickly reviewing the basics of integration on a measure space, although we will then translate the key results of this theory into probabilistic language.
As the finer details of the Lebesgue integral construction are not the core focus of this probability course, some of the details of this construction will be left to exercises. See also Chapter 1 of Durrett, or these previous blog notes, for a more detailed treatment.


text:
|Height: 6-0 | Weight: 250 | College: Citadel
Positives Has a thick, solid frame with a strong, muscular build, big bubble, thick thighs and calves. Has good initial quickness and foot speed, showing enough acceleration to clear the line of scrimmage. Plays with good balance and is able to keep his feet while pushing the pile. Has adequate change of direction agility to make upfield cuts. Strong downhill runner who will create his own holes with his strength when he cannot find a free lane. Has the body lean and forward push to be much more effective between tackles than on the outside. Plays at a proper pad level, maintaining balance through his run. The type of runner who can move the chains back and consistently gains yards after initial contact. Has the upper body strength to run through arm tackles. Has the speed to get open up a seam in the passing game. Looks natural catching the ball, but is better in the short area than on deep routes (hip stiffness). Shows the ability to catch outside the body's frame. Takes on defenders with force as a blocker.
Negatives Has good field awareness and will retain plays, but does not always anticipate defensive coverage or see the play develop. Will miss the cut back or run through the wrong holes at times. Lacks the hip swerve and lateral agility to bounce outside. Has the power to break tackles, but will get brought down by low tackles trying to navigate through trash. Needs to show more aggression in his play. Lacks the second gear needed to pull away from defenders in the open. Lack of flexibility makes him struggle trying to get into gear on long routes and is better when catching in the short area. Has problems moving his feet and sliding in pass protection, lacking proper hand technique to gain leverage as a blocker.
|2004 Season
Started the first ten games at tailback On a second quarter run in the Chattanooga contest, his leg gave out and had to be carted off the field Tests revealed a torn anterior cruciate ligament in his left knee Underwent reconstructive surgery, missing the final two games vs. Virginia Military Institute and East Tennessee State Still managed to lead the team with 778 yards on 165 carries (4.7 avg) and eight touchdowns Third on the team with 21 catches for 239 yards (11.4 avg) and two scores Also made a solo tackle Gained 1,017 all-purpose yards Did not fumble the ball and lost only 47 yards on his 165 rushing attempts.
2003 GAME Analysis
Delaware Made a solo tackle and gained 37 yards on 12 runs, capping a 12-play, 87-yard drive in the third quarter with a 3-yard touchdown burst.
Western Carolina Rushed 22 times for 57 yards and a 1-yard score that concluded an 80-yard, 10-play drive in the first quarter Also snared four passes for 67 yards.
Appalachian State Followed with eight attempts for 59 yards and a 33-yard score.
Elon Piled up 147 yards on 14 chances with three touchdowns and caught four passes for 51 yards.
Furman Carried 28 times for 159 yards and gained 14 yards on a pair of grabs.
Georgia Southern Picked up 127 yards on 28 tries with a pair of scores and made three catches for 9 yards.
Chattanooga Ran 11 times for 50 yards and had touchdown receptions of 18 and 35 yards on tosses from QB Willie Simmons On a first-&-10 run from the Chattanooga 37 in the second quarter, Nemo suffered what he first thought was an ankle sprain, but tests in the locker room revealed a torn ACL in his left knee, finishing his season.
|2003 Season
All-Southern Conference second-team choice Played in every game, starting five contests at tailback and two more at fullback Led the team with a career-high 1038 yards and eleven touchdowns on 224 carries (4.6 avg) Caught 18 passes for 181 yards (10.1 avg) and a score Added 32 yards on a pair of kickoff returns Also made two tackles
Fumbled the ball four times, with the opposition turning three of those miscues into scores Lost 45 yards on his 224 rushing attempts Ran for over 100 yards in six contests.
Louisiana State Capped a 14-play, 81-yard drive in the third quarter with a 1-yard touch-down run, gaining 15 yards on six carries, 6 yards on a catch and also made an assisted tackle.
Western Carolina Had the first 100-yard game of his career, totaling 121 yards on 14 attempts and 32 yards on two kickoff returns.
Appalachian State Turned the ball over once on a fumble, picking up 100 yards on 24 tries and 60 yards on five receptions.
Wyoming Turned the ball over again on a fumble, but collected a season-high 175 yards on 33 carries and 52 yards on five catches.
East Tennessee Recorded a solo tackle, but also had another fumble for the third-straight game Finished with 19 runs for 99 yards and touchdowns of 2 & 12 yards.
Georgia Southern Added20 rushes for 109 yards and a touchdown.
Chattanooga Had 28 carries for 88 yards and a touchdown.
Virginia Military Institute Followed with 30 runs for 111 yards and a score, adding 30 yards on three receptions.
Charleston Southern Closed out the season with 32 yards on three catches and 109 yards with a touchdown on 22 carries.
|2002 Season
Appeared in eight games, seeing action mostly as a blocking back Carried 13 times for 34 yards (2.6 avg) and a touchdown Ran three times for 11 yards and a 2-yard score vs. South Carolina State Followed with five runs for 14 yards vs. East Tennessee.
|Injury Report
1999: Suffered a torn anterior cruciate ligament in his left knee as a junior in high school. He did not have surgery and the ligament would again tear in 2003.
2003: Sat out the East Tennessee and Virginia Military Institute games after tearing the ACL in his left knee in the second quarter of the Chattanooga game (11/08). Nehemiah underwent reconstructive surgery to repair his knee on Nov. 21 Was held out of 2004 preseason drills to continue his rehabilitation.
Business Administration/Sports Management major Son of Sandra Broughton and Nehemiah Broughton Sr .Born 11/04/82 Resides in North Charleston, South Carolina.
|High School
CAMPUS: 4.63 in the 40-yard dash 410-pound bench press 670-pound squat 340-pound power clean Right-handed 15/21 Wonderlic score. COMBINE: 4.61 in the 40-yard dash 2.7 20-yard dash 1.61 10-yard dash 4.28 20-yard shuttle 11.91 60-yard shuttle 7.54 three-cone drill 30-inch vertical jump 9'0" broad jump Bench-pressed 225 pounds 25 times 32 3/8-inch arm length 9 4/4-inch hands 18 Wonderlic score.
Attended North Charleston (S.C.) High School, playing football for coach Nelson Huggins All-State and All-Low Country selection as a senior Rushed for 1,670 yards and 23 touchdowns during his final campaign and played in the North-South All-Star game.
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About Brian
Meet Brian Harnish – Developer, Strategist, Speaker, Author
Aside from the usual traits of being able to build web sites from scratch using the core markup & styling languages HTML & CSS (and graphics talent with Adobe Photoshop), Brian has spent the last 17 years in website design & development, spending the last 8 years as an SEO professional.
So when and how did this whole obsession with SEO start? This actually started in the spring of 1998, the year he graduated from high school. He started learning how to code. In the summer, he took a college level Photoshop class and the rest as you know it was history. He spent his time from then on learning the intricacies of graphics and coding, to the point that he can now code an entire website from scratch if he needs to. Obviously, some tools are now in place to help speed up productivity and add efficiency where desired.
How Brian Accidentally Stumbled Into the Industry — in 2001
One of Brian's first clients was Cleavant Derricks, an actor, singer & performer who just so happened to be Rembrandt "Crying Man" Brown on the immensely popular Fox TV show Sliders. Brian first met Cleavant via a referral from another existing client who knew him, and the rest is pretty much history. Since 2001 Brian has proudly hosted & monitored the upkeep & maintenance of Cleavant's official website with the occasional updates (the latest design update is on hold for the foreseeable future). Brian has had a number of clients through the years. However, not every website is still up!
After a Succession of Minor Successes Later, Brian Became a Professional SEO
In 2006 Brian got a dream opportunity – the opportunity to start at a support department in a real estate website design & hosting agency named Advanced Access. Here Brian was able to expand his technical problem solving abilities and develop client relationships. He was promoted several times, and eventually got promoted into the Internet Marketing Department in 2007. Here is where the internet marketing industry just randomly found him.
From 2010 until recently Brian was able to serve as an Internet Marketing Specialist for Bisnar | Chase Personal Injury Attorneys, LLP. As part of their in house marketing team, Brian was able to realize a number of fantastic achievements. He was accepted to speak at conferences worldwide including: SMX Toronto 2012, SMX East 2012, and SMX West 2013. Brian was invited back to SMX West 2014 but was, unfortunately, unable to attend. In addition, Brian contributed significantly to taking BestAttorney.com from 7,000 visitors/mo to an astonishing 15,000 visitors/mo. In addition to the obvious contributions, Brian was also able to expand his skill set significantly, adding PHP and updated DIV/CSS layout capabilities. His philosophy is "if you aren't learning, you're stagnating. So keep learning!"
Philanthropic Commitments & Responsibilities
In 2010 Brian got to realize a fantastic opportunity by becoming an organizer for We're Going Back, an organization of Back to the Future fans working under the Michael J. Fox Foundation for Parkinson's Research. They setup a full Back to the Future 30th anniversary reunion. Brian was able to organize the world's first Twitter cast of the entire Back to the Future trilogy, set to coincide the week before We're Going Back launched the week of November 5, 2010 (for those who don't know, Nov. 5 – 12 were the dates Marty was sent back to 1955 in the movie). As a result, Brian was able to help raise $100,000.00 for the Michael J. Fox Foundation for Parkinson's Research.
Brian has been able to repeat his 2010 successes by being hired back to help with social media promotion of the 2015 30th anniversary We're Going Back Back to the Future Fan Celebrations. For Brian, it is a dream come true to be on the inside of one of the nation's most obsessed group of Back to the Future fans.
Brian attended Orange Coast College and finished in 2009 with a certificate in website design.
In his spare time, you can typically find Brian obsessing about DeLoreans, listening to music (mostly Elvis Presley), playing video games, watching movies, and obsessing about SEO on Facebook & Twitter.


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Running Time: 148 minutes
Released: 08/2008
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Jethro won it for Labour
Posted - 20 April 2017 08:18
Put an advert in the local advertiser?
"Indecent ugly fantasist, 50, seeks desperate people to be his friends. Will pay handsomely".
O. Radiant Dawn
Posted - 21 April 2017 06:14
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I was thinking about something involving balloons and cake actually.
Posted - 21 April 2017 06:32
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drive up to dubai and spaff out in crown plaza air hostesses bar
Kaylee Frye
Posted - 21 April 2017 07:48
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Stop posting on RoF for a year.
Posted - 21 April 2017 07:54
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to hijack the thread - and apropos of nothing - the day after my 50th birthday, my daughter is 21.
will we have a family party? will she want to hang out with me; she'll have just finished uni. Should I start planning now?
Posted - 21 April 2017 08:06
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Buy a sex robot Hugely.
Posted - 21 April 2017 08:07
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You could make a weekend of it.
Hire a country house or something, your party on Friday and hers on Saturday, people starying over.
Start saving!
Posted - 21 April 2017 08:23
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that's an idea - could do the whole country house thing....
Weally Been
Posted - 21 April 2017 08:37
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On my 50th birthday my daughter will have just turned 16 so life will be quite different for me then I can imagine
Posted - 21 April 2017 09:43
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wtf i thought you were in your 30s from the stuff you post.
🐝 buzz
Posted - 21 April 2017 09:48
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Well his former email address appears on an internet dating advert published in 2000 stating he's 32 so do the mathS. Yes, maths has an "s" at the end.
Posted - 21 April 2017 09:49
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blasphemy, buzz
Posted - 21 April 2017 09:49
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Depends if you like parties and want to waste the money. I chose not to.
Jethro won it for Labour
Posted - 21 April 2017 09:51
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Because you are the only person on the planet who prefers sitting in her bedroom alone than going out and getting pissed with their mates and having a laugh Lydia.
🐝 buzz
Posted - 21 April 2017 09:52
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That's because you're a joyless antisocial materialistic miserabalist.
ReggiePerrin is voting Lib Dem
Posted - 21 April 2017 09:56
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Wot Buzz said about maths.
Posted - 21 April 2017 10:05
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Of course we are assuming here that hyoobert told the truth in his lonely hearts ad.
Given what one knows of those publications and also poor little hyoobert's relationship with veracity we really shouldn't discount the possibility that his half century has actually been and gone
Posted - 21 April 2017 10:07
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I don't fucking trawl dating sites checking up on roffers
Posted - 21 April 2017 10:11
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was it a dating site? thought it was porn
Posted - 21 April 2017 12:45
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I'm very happy and secondly I really don't think anyone could easily call me a materialist.
My brother had a 50th party last year - I went but it was his wife who wanted (they are virtually the same agem, she is very slightly older and it was joint).
If you want one have one.


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Sometimes, you may be confused to choose which one is the best lip balm for dark lips because you will find thousands of lip balms in the cosmetics shops. When you buy a lip balm, you should think about some particular things such as- which ingredients are used to make balms- natural or outside elements, whether it is useful for all seasons, whether it contains sun or wind protection, whether it has nutrients to make the lip skin healthy, whether it is safe for skin etc.
Anyway, today I will show you some best lip balm for dark lips.
Read Also: Natural Remedies for Dark Lips
The Best Lip Balm/Chapstick for Dark Lips
During choosing best lip balm for dark lips, you have to consider a few of things, like a good price, natural ingredients, safety for the lips, leading brand and nice customer experiences. If all these things are like that, your lip balms can be good and treat your dark lips perfectly.
Okay, let's see the best lip balm for dark lips.
1. Lalune Naturals Organic Lip Balm
This lip balm is made with natural organic flavors, organic Calendula, organic rosemary extract, vitamin E, organic coconut oil, organic beeswax, and organic sunflower oil. Lalune Naturals Lip Balm is safe for all ages and usable for cracked, dry, chapped or dark lips. You will get 4 natural flavors. It is not tested on the animals and not added any soy, fillers, petroleum derivatives, chemicals, gluten, GMO, and toxic.
2. EOS Organic Lip Balm Stick in Vanilla Bean
EOS Organic Lip Balm is a long-lasting formula for dry, chapped, cracked or dark lips. It's Shea butter and vitamin E protect your lips from drying up and moisturize them all day long. If dryness causes your lips dark, EOS lip balm is a good solution for you.
3. Kiss My Face Sports Lip Balm with SPF 30
The lips of athletes get burned, cracked and black by huge sun exposure. Kiss My Face Sports Lip Balm SPF 30 is a nice chapstick for them. It safeguards your lips during play against the odd situations. It is made with organic beeswax, organic coconut oil, antioxidants, Vitamin E, and Shea butter that soften, soothe and moisturize your lips. Kiss My Face Sports Lip balm is also FDA approved protector for the sun rays. Both adults and children can apply it. It is water resistant, natural flavor rich, and has no preservatives.
4. Rosebud Lip Balm in Minted Rose
Rosebud Lip Balm, Minted Rose is an original ointment for all skin purposes, like diaper rash, blemishes, chapped lips, rough cuticles, sun/wind burn and detergent burns. It soothes, moisturizes and softens the damaged lips. You can also use Rosebud lip balm to soften your rough hand skin.
5. Neutrogena Norwegian Formula Lip Moisturizer SPF 15 Lip Balm
Neutrogena Norwegian Formula Lip Moisturizer Sunscreen, SPF 15 Lip Balm is a dermatologist recommended product for dry, chapped, cracked, burned, dark and damaged lips. It is fragrance and PABA free, and no waxy feel. Protecting your lips from sunburn, this balm keeps your lips conditioning, soft and moisturized.
6. Burt's Bees 100% Natural Beeswax Lip Balm (Full Review)
Burt's Bees lip balm is made with 100% natural ingredients. The natural ingredients are beeswax, vitamin E, peppermint oils. These ingredients make your dark lips bright, pink, conditioning, smooth, refresh and moisturize.
- 100% natural
- 15-ounce tubes
- Natural lip-conditioning with original beeswax
- Healthy and smooth lips with antioxidant vitamin E
- Refresh glowing lips with peppermint oilBuy it at amazon.com
7. Body Shop Watermelon Born Lippy Balm
Body Shop Watermelon Born Lippy Balm is very expensive, so best lip balm for dark lips. You will be really astonished when you will apply it on your lips. It removes the dark tones from the lips. It also rejuvenates the outlook of your lips that adds extra beauty to your appearance. It contains Shea butter, pineapple, and strawberry that are very helpful to lighten the dark lips.Buy it at amazon.com
8. Lip Smackers Coca Cola Fanta Sprite Coke Bargs Lip Balm
Lip Smackers Coca Cola Fanta is an effective lip balm for black lips. It is infused with different flavors that spread the lips' attractiveness. It is also great for dry or chapped lips. It hydrates and keeps the lips mild. You can easily apply this lip balm for any party function because it increases the glamor of lips.
- Flavors: coca- cola- coca- cola cherry- coca- cola vanilla- sprite- barq's root beer- fanta grape- fanta orange- fanta strawberry
- Lip smacker Coca-Cola party packs 8 count
- Give 'em, Collect 'em, Share 'em
- Great tasting coca cola lip treats… Perfect for party fun!
- Smooth and moisturized lips with one touch!Buy it at amazon.com
9. Lotus Herbals Lip Balm in Raspberry
Lotus Herbals Lip Balm Raspberry is one of the best lip balms for dark lips because of having the wax like texture. It stops the growing of darkness on the lips. Lotus Herbal Lip Balm keeps your lips hydrated, healthy and soft. You will get six different sweet colors like vanilla, cocoa, fresh mint, cherry and chocolate. It is a costly product, but very effective, so very popular also.
10. Maybelline New York Baby Lips Moisturizing Lip Balm in Cherry Me (Full Review)
Maybelline lip balm is very popular with the women having black lips. It is developed with SPF 20 sunscreen to protect the lips from UV damages. It also helps to renew the cells so that the lips look younger. It keeps the lips moisturized for 8 hours and tightens the skin so that you feel a tender touch on your dark lips.
- Moisturize up to 8 hours
- SPF 20 sunscreen
- Skin on lip renewal
- Clinically strengthBuy it at amazon.com
11. Bonne Bell Lip Smacker Party Pack Flavored Lip Balm Skittles
Bonne Bell Lip Smacker Party Pack Flavored Lip Balm is favorable to many women having dark or black lips because this lip balm is a great fun for party gatherings. It has gorgeously sweet flavors that attract your friends at party fun. It enhances your lip's beauty and makes the lips very tender with a hot look. You can easily smile with this funny lip balm for 6-8 hours.
- Start your party with it!
- Great lasting lip balm
- Super moisturizing and healing properties from damages
- Fantastic flavors lip SmackersBuy it at amazon.com
12. Jack Black Intense Therapy Lip Balm SPF 25
Jack Black Intense is one of the best lip balms for dark lips because It won the Best Lip Balm, Best of Allure, 2012 award. It protects your dark or dry lips from the wind and the sun. This lip balm has some effective moisturizers and antioxidants that treat your lips perfectly. It is made with SPF 25 that give you instant protection from sun rays.
13. Beauty by Earth Peppermint 100% Natural Beeswax Lip Balm
Beauty by Earth Lip Balm is a great lip balm for black and dry lips. It is made with various natural and effective to skin ingredients like- cocoa butter, vitamin E, Shea butter, beeswax, and peppermint. It has no any toxic substances. It is better because of repairing and moisturizing the dry and dark lips. You will feel longer lasting affection after applying this lip balm.
- 100% natural remedies- sunflower seed oil, beeswax, castor seed oil, lanolin, vitamin E (non-GMO), peppermint, coconut oil, Aloe Vera, Shea butter, and cocoa seed butter.
- Non-GMO and non-toxic: Nothing toxic, no corn, no petroleum or petroleum derivatives, no chemicals, no soy, no cheap fillers, and no junk!
- Moisturize utterly
- Money back guarantee
- Made in the U.S.ABuy it at amazon.com
14. Rosebud Three Lavish Layers Lip Balm
Rosebud Three Lavish Layers Lip Balm is one of the best lip balms both for dark and chapped lips. You can use it either winter or any other season. It makes your dark lips moisturized and smooth for a long time. It also protects your lips in winter from being dry or chapped.
15. EOS Lip Balm
EOS Lip Balm is especially good for dark lips because it is made with some fantastic natural ingredients. The ingredients are blueberry, honeysuckle honeydew, sweet mint etc. All these ingredients will take care of your lips with a natural touch that presents you a pair of healthy, beautiful, and bright and attractive lips. This lip balm also protects your lips from being dark or dry. No doubt, you can apply it to get an impressive lip look at all times.
- Honeysuckle honeydew, tangerine, summer fruit, lemon drop, sweet mint, blueberry, pomegranate, and strawberry sorbet
- 8 different wrapped lip balms
- Pucker up. Smooth on. Twist off the top.Buy it at amazon.com
16. Nivea Lip Care A Kiss of Olive and Lemon Lip Balm
Nivea is a famous brand name in the world for cosmetics. Having SPF 15, it also saves your lips from various damages. It keeps your lips highly soft and smooth with retaining the real color. It contains a mildly sweet fragrance that spreads your beauty to others. No doubt, Nivea is one of the best lip balms for dark lips.
17. Neutrogena Naturals Lip Balm
It is a vitamin E enriched bio-nutrient best lip balm for dark lips. Vitamin E soothes, moisturizes and nourishes the lips. It is free from any harmful ingredients like petrochemicals, sulfates etc. The bio-nutrients helps to bring back the natural look of your lips.
- Vitamin E bio-nutrient lip balm
- Dermatologist recommended brand
- Nourishes and soothes lip
- Essential moisturizers
- No sulfates, petrochemicals, phthalates, dyes, or parabens
- Bio-nutrients work naturallyBuy it at amazon.com
18. Tony Moly Blueberry Mini Lip Balm (Full Review)
The most charming thing about Tony Moly Blueberry Lip Balm is its container that is designed with a real blueberry shape. If you did not see it before, you would think that it is a genuine blueberry. Most of the customers are very craving for its sweet container. The core balm inside the pot is also very effective for dark or dry lips. It leaves a nice red tint that looks the ripe blueberry color and the tint is so natural that nobody can understand it is a balm. Tony Moly smells as like as raw berry. No artificial scent is added. It is long lasting, super moisturizing, hydrating and makes the lips shiny and glossy, but not greasy or sticky.Buy it at amazon.com
19. Burt's Bees Ultra Conditioning Lip Balm with Kokum Butter
Again, get an another natural lip balm for your black lips. It is formulated with cocoa, Kokum, and Shea butter. Just with one touch, you will get a pair of healthy, supple and soft lips. It keeps your lips moisturized for 4 hours with just a touch. It is made with no fragrances or flavors. You can be free from any allergic reaction. A natural look will be created on your lips that make your lips hot and sexy.
- Moisturizing lip for 4 hours
- No flavors or fragrances
- Hypoallergenic
- Replenishing lips with Kokum, cocoa, and Shea butter
- Soft, healthy and supple feeling on lips
- Pack of 4, 100% naturalBuy it at amazon.com
20. Avon Lip Balm
It comes with 3 different flavors. This group lip balms are very popular with the tender aged girls. It is very effective both for dark and dry lips. Men are also applying this lip balm. It gives bright color and gloss. It also helps to heal the lips from sun and wind damaging.
It is time to remove the darkness on your lips as you have already got your best lip balm for your dark lips. We always recommend not to use the so-called and cheap lip products, because they sometimes contain harmful elements. To get a good safety for your lips, you can use lip balms with SPF sunscreen.
If you like the above best lip balm for dark lips, don't forget to share it with others.


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Sunday Birth Shop
Florida Hospital Wesley Chapel
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This is a one day quick course that teaches expectant moms about labor/delivery, medications, epidural, breathing/relaxation, Cesarean section and induction.
In order to purchase these tickets in installments, you'll need an Eventbrite account. Log in or sign up for a free account to continue.


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10 properties in Zvolen
Zvolen Accommodation
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Slovakia   accommodation in Zvolen - Hotels.com offers great last-minute Zvolen hotels deals and discount room rates. Our hotels in Zvolen can easily be booked online. Our fast online search facility allows you to:
- - Find and compare different hotels in Zvolen by star rating, amenities, theme, or brand
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At Garden Express you can make a purchase with confidence. Our living merchandise is normally guaranteed to grow if our free growing guide instructions are followed. Any guaranteed items that do not grow within 3 months (6 months for dormant items) will be replaced, credited to your account or fully refunded.
The exception to this guarantee is discounted, end of season stock which is sold on an 'as is' basis. Generally this stock is sound but should be planted as soon as possible and may not perform as well as normal in season products.
Delivery is normally via Australia Post with larger orders delivered via StarTrack Express* courier. We service virtually all areas of Australia with only those areas not serviced by these organisations being excluded (e.g. remote islands and Territories such as Christmas Island, Lord Howe Island, Norfolk Island, Cocos Islands, Groote Eylandt). We do not deliver outside Australia but happily deliver orders placed from overseas for Australian delivery. Our regular delivery charge for basic items is $9.50 Australia-wide, regardless of the number of items ordered. Due to their higher freight costs, bulky or heavy products such as roses, large pots and a small number of hard goods have higher delivery charges. The summary of all current delivery options is:
- Orders NOT containing Roses, 120mm and larger pots/grow-bags, Bare-rooted Trees, Composting Buckets, nor any items with individually specified delivery charges, all delivered at one time, will incur the normal $9.50 freight charge. These items are referred to as "Basic Items".
- Orders containing bush Roses, 120mm up to 155mm pot/grow-bags, Composting Buckets, together with other products delivered at the same time will incur a $13.90 freight charge.
- Orders containing Standard and/or Patio Roses, 160mm up to a 200mm pot/grow-bags, Bare-rooted Trees, together with other products delivered at the same time will incur a $20.00 freight charge.
- Orders containing Weeping Roses, Garden Beds, Greenhouses or pots/grow-bags greater than 200mm, together with other products delivered at the same time, will incur a $30.00 freight charge.
- Orders containing products with explicitly stated delivery charges will incur charges as detailed in the individual product description.
Orders containing products from two different seasons (eg Rose or spring bulb pre-orders) that require separate deliveries will incur two delivery charges at the relevant rates (unless a free freight offer applies). A $4.00 Quarantine Inspection Surcharge applies in addition to the Delivery Charge for orders delivered to Tasmania, Northern Territory and Western Australia due to the increased complexity of quarantine requirements.
As quarantine inspections apply to each delivery, 2 quarantine charges will apply where orders contain products from different seasons. *Please note that if you order large items (eg Patio, Standard or Weeping Roses, Fruit Trees) you must specify a street address for the delivery address. StarTrack Express generally do not deliver to PO Boxes or RSD addresses so your order will be held until a suitable address is obtained. In rural areas StarTrack may deliver to physical addresses such as general stores that act as Australia Post agencies but we suggest you check with them first. Where 'Free Delivery' vouchers apply to an order, details of the voucher should be entered in the "Your message to us" field. Orders (except Pre-orders) are normally despatched within 7-14 working days from the time of ordering during busy periods. Delivery times may be extended during peak periods (typically following the release of the latest catalogue), extended public holidays, if extreme weather conditions are not suitable for live plant transport or as detailed in the product description. Individual items may be back ordered to avoid delaying the delivery of other items. Please allow additional delivery time for orders to Tasmania, Northern Territory and Western Australia due to quarantine restrictions.
Please note: Some orders may not be possible to parts of Australia due to quarantine and some items may carry unique delivery charges. Check full delivery policy.
Common Name: HYDRANGEA Also known as Mopheap Hydrangea
Botanical Name: Hydrangea macrophylla
Family Name: Saxifragraceae
Plant Type: Deciduous shrub
Size: 0.5–3m H x 0.5–2m W
Flowering: Summer to Autumn
Positioning: Semi Shade to Full Shade
General Information: Hydrangeas thrive in a cool, moist position, sheltered from hot summer sun and northerly winds.
Soil: Moist, rich, well-drained.
Watering: Regular watering is required, especially in late spring and through summer. Deep soaking once or twice a week is ideal. Have a saucer of water underneath pots.
Fertilising: In winter while the plant is dormant, use a fertiliser designed for the pH (colour) you are after. Also fertilise in spring.
Pruning: Prune back by about 1/3 when dormant in winter. Cut back to double leaf buds to encourage flowers the following year.
Tips: Flower colours can be enhanced or altered depending on the pH level in the soil low pH makes them blue, high pH makes them pink.


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1 Store
Parchment Paper CertificatesAcid-free, 60-lb. parchment paper certificates have border design. Can be used with copiers, laser and inkjet printers. Each certificate is made from 50 Percent recycled, 30 Percent post-consumer material and can be decorated with self-adhesive gold foil seals (sold separately). For use with Geographics document covers. Theme: Royalty; Shape: Horizontal Rectangle; Width: 11 .


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I'm writing from Switzerland, heaven on earth for hikers and cyclers. Walking and riding bicycles is a way of life in Switzerland. There are no school buses – children walk or ride bicycles to school. Senior citizens walk to the store to a very ripe old age or you might see them riding a bike just for fun.
We've had some good backpacking recipes submitted this month.
Click to view these new backpacking recipes.
I'll be back in the US in early May. I'm hiking the last 100 miles or so of the Appalachian Trail in Virginia this spring. Maybe I will see you out there.
Best regards,
Chef Glenn at BackpackingChef.com


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- Description
Finally, a true high current, audiophile quality, power distribution system.
The CMX-6 provides clean power, free of high-frequency line noise, to six commercial grade outlets – enabling your audio components to deliver their best performance under all operating conditions. Its high current construction is fully rated for use with power hungry amplifiers and other high performance audio components. It also continuously monitors your AC powerline wiring integrity and alerts you to common fault conditions like open grounds and reversed lines.
- Ideal accessory for high performance electronics.
- Fully rated for use with high power amplifiers, will not limit peak current.
- High-quality common mode and differential mode L-C noise filtering.
- Continuous line-status monitoring - provides continuous real-time indicators that show the status of Hot, Neutral, and Ground lines.
- Power Direct Connect™ mounting system, which connects the outlets directly to the circuit board.
- Premium high-quality components and a heavy-duty double sided FR4 circuit board with high current paths on both sides.
- Six commercial grade AC outlets.
- Detachable IEC high current, 14 gauge, 2-meter power cord included.
- High reliability resettable magnetic circuit breaker.
- Constructed from machined aluminum.
- Can be mounted vertically or horizontally.
AC Power Source:
120 VAC +/- 20% @ 60 Hz; 15 Amps
Load (total for all outlets):
15 amps continuous precision magnetic circuit breaker allows for short-term inrush current of up to 250% of rated load
Power cable:
Standard IEC detachable power cord; 7 feet long; 14 AWG x 3
unboxed: 4" wide x 1 3/4" high x 16 3/4" deep
boxed: 9" wide x 4 1/4" high x 20 1/2" deep
- Weight: 3.9 lbs (5.6 lbs boxed)


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Sophisticated and simple, this flat ballerina in super-soft nubuck leather can be worn for almost any occasion. With a fashionable pointed toe, it will keep your look up to date, while features designed for comfort help ensure relaxed walking all day long.
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At ECCO we are committed to make your online shopping experience as easy as possible. The best way to make sure that you order the right size is to measure your feet and then compare it with the size chart below to find the appropriate size. Please follow these 4 simple steps to accurately measure your feet:
All your need to measure your feet from heel to toe is a piece of paper, a pencil and a ruler.
Place the paper on the floor up against a wall. Place the heel against the wall.
Mark the longest part of your foot on the paper. Repeat procedure with the other foot.
Measure the length of each mark and match it with the size chart.
Let the longest foot decide the shoe size.
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