theorem The central limit theorem states that the sum of a number of independent and identically distributed random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with a power-law tail (Paretian tail) distributions decreasing as where (and therefore having infinite variance) will tend to a stable distribution as the number of summands grows. If then the sum converges to a stable distribution with stability parameter equal to 2, i.e. a Gaussian distribution. Dependent processes CLT under weak dependence A useful generalization of a sequence of independent, identically distributed random variables is a mixing random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially strong mixing (also called α-mixing) defined by where is so-called strong mixing coefficient. A simplified formulation of the central limit theorem under strong mixing is: In fact, where the series converges absolutely. The assumption cannot be omitted, since the asymptotic normality fails for where are another stationary sequence. There is a stronger version of the theorem: the assumption is replaced with and the assumption is replaced with Existence of such ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see . Martingale difference CLT Remarks Proof of classical CLT The central limit theorem has a proof using characteristic functions. It is similar to the proof of the (weak) law of large numbers. Assume are independent and identically distributed random variables, each with mean and finite variance The sum has mean and variance Consider the random variable where in the last step we defined the new random variables each with zero mean and unit variance The characteristic function of is given by where in the last step we used the fact that all of the are identically distributed. The characteristic function of is, by Taylor's theorem, where is "little notation" for some function of that goes to zero more rapidly than By the limit of the exponential function the characteristic function of equals All of the higher order terms vanish in the limit The right hand side equals the characteristic function of a standard normal distribution , which implies through Lévy's continuity theorem that the distribution