Infinitely Divisible Laws

Abstract

It is a remarkable fact that the class of limit distributions of normed sums of i.i.d. random variables is severely circumscribed. If the underlying r.v.s, say {X _n , n ≥ 1} have merely absolute moments of order r, then for r ≥ 2 only the normal distribution can arise as a limit, while if 0 < r ≤ 2, the limit law belongs to a class called stable distributions. If the basic r.v.s are merely independent (and infinitesimal when normed cf. (1) of Section 2), a larger class of limit laws, the so-called class ℒ emerges. But even the class ℒ does not contain a distribution of such crucial importance as the Poisson. A perusal of the derivation (Chapter 2) of the Poisson law as a limit of binomial laws B _n reveals that the success probability associated with B _n is a function of n. Thus, if B _n-1 is envisaged as the distribution of the sum of i.i.d. random variables Y _l, …, Y _n-1, then B _n must be the distribution of the sum of n different i.i.d. random variables which, therefore, may as well be labeled X _{n, 1}, X _{n, 2},..., X _{n, n} .