San José State University

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 Infinitesimal Random Variables

In the body of mathematical analysis leading up to stable distributions there are several concepts that are crucial. Infinitely divisible random variables is one of those concepts and infinitesimal random variables is another. Infinitely divisible random variables are covered elsewhere. The purpose of this page is to present some of that mathematical analysis involved in the concept of infinitesimal random variables and to present an important lemma discovered by Ya. Khintchine.

The idea involved in infinitesimal random variables is that the limit distributions of sums of an increasing number of independent random variables should be such that the influence of any individual summand should become vanishingly small as the number of summand increases without bound.

The precise definition of infinitesimal random variable is given in terms of a construction involved in getting a limit distribution. Let z1, z2, ... , zn, ... be a sequence of independent random variables such that any one variable is the sum of a certain number of mutually independent random variables; i.e., for all n

#### zn = un1 + un2 + ... + unkn.

The set of random variables {{un1, un2, unkn} for n=1, 2, ...} is infinitesimal if for any ε > 0

#### limn->∞[ sup1≤k≤kn Prob{|unk|≥ε}] = 0

where sup stands for supremum, the least upper bound.

Let Pnk(u) be the cumulative probability distribution of the random variable unk. Then the following lemma holds:

## lim Sn = 0 as n→∞.

(To be continued.)

Source:

• B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions of Sums of Independent Random Variables (translated by K.L. Chung), Addison-Wesley Publishing Co., Cambridge, MA, 1954.