Kolmogorov's three-series theorem
Introduction In probability theory, '''Kolmogorov's Three-Series Theorem''', named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions. Kolmogorov's three-series theorem, combined with Kronecker's lemma, can be used to give a relatively easy proof of the Strong Law of Large Numbers.
Statement of the theorem Let  be independent random variables. The random series  converges almost surely in  if the following conditions hold for some , and only if the following conditions hold for any :
Proof === Sufficiency of conditions ("if") ===Condition (i) and Borel–Cantelli give that  for  large, almost surely. Hence  converges if and only if  converges. Conditions (ii)-(iii) and Kolmogorov's Two-Series Theorem give the almost sure convergence of .=== Necessity of conditions ("only if") ===Suppose that  converges almost surely.Without condition (i), by Borel–Cantelli there would exist some  such that  for infinitely many , almost surely. But then the series would diverge. Therefore, we must have condition (i).We see that condition (iii) implies condition (ii): Kolmogorov's two-series theorem along with condition (i) applied to the case  gives the convergence of . So given the convergence of , we have   converges, so condition (ii) is implied.Thus, it only remains to demonstrate the necessity of condition (iii), and we will have obtained the full result. It is equivalent to check condition (iii) for the series where for each ,  and  are IID—that is, to employ the assumption that , since  is a sequence of random variables bounded by 2, converging almost surely, and with . So we wish to check that if  converges, then  converges as well. This is a special case of a more general result from martingale theory with summands equal to the increments of a martingale sequence and the same conditions (; the series of the variances is converging; and the summands are bounded).
Example As an illustration of the theorem, consider the example of the ''harmonic series with random signs'':: Here, "" means that each term  is taken with a random sign that is either  or  with respective probabilities , and all random signs are chosen independently. Let  in the theorem denote a random variable that takes the values  and  with equal probabilities. With  the summands of the first two series are identically zero and var(Yn)=. The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely. On the other hand, the analogous series of (for example) square root reciprocals with random signs, namely: ''diverges'' almost surely, since condition (3) in the theorem is not satisfied for any A. Note that this is different from the behavior of the analogous series with ''alternating'' signs, , which does converge.
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