t′ goes to u (as n increases) when In other words, the characteristic function converges pointwise to and therefore by Lévy's continuity theorem the sum divided by converges in distribution to the symmetric alpha-stable distribution with stability parameter and scale parameter 1. This can be applied to a random variable whose tails decrease as . This random variable has a mean but the variance is infinite. Let us take the following distribution: We can write this as where We want to find the leading terms of the asymptotic expansion of the characteristic function. The characteristic function of the probability distribution is so the characteristic function for f(x) is and we can calculate: where and are constants. Therefore, and according to what was said above (and the fact that the variance of f(x;2,0,1,0) is 2), the sum of n instances of this random variable, divided by will converge in distribution to a Gaussian distribution with variance 1. But the variance at any particular n will still be infinite. Note that the width of the limiting distribution grows faster than in the case where the random variable has a finite variance (in which case the width grows as the square root of n). The average, obtained by dividing the sum by n, tends toward a Gaussian whose width approaches zero as n increases, in accordance with the Law of large numbers. Special cases There is no general analytic solution for the form of f(x). There are, however three special cases which can be expressed in terms of elementary functions as can be seen by inspection of the characteristic function: For the distribution reduces to a Gaussian distribution with variance σ2 = 2c2 and mean μ; the skewness parameter has no effect. For and the distribution reduces to a Cauchy distribution with scale parameter c and shift parameter μ. For and the distribution reduces to a Lévy distribution with scale parameter c and shift parameter μ. Note that the above three distributions are also connected, in the following way: A standard Cauchy random variable can be viewed as a mixture of Gaussian random variables (all with mean zero), with the variance being drawn from a standard Lévy distribution. And in fact this is a special case of a more general theorem (See p. 59 of ) which allows any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter of