San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

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:

Lemma: The set of random variables
{unk, k=1,..,kn; n=1, 2, ..}
is infinitesimal if and only if
for any ε > 0

limn->∞[sup1≤k≤kn ∫{(u2/(1+u2)dPnk(u)}] = 0

Necessary and Sufficient Conditions that
a Set of Random Variables Are Infinitesimal

Lemma: Let {ζn, n=1, 2, ...} be a sequence of random variables such that each is the sum of independent random variables; i.e., ζn = ζn1 + ζn2 +...+ ζnkn. Let Fnk be the cumulative distribution function of the variable ζn. Let Enk be the expected value of of x2/(1+x2) with respect to the cumulative distribution function Fnk; i.e.,

Let Sn be the supremum (least upper bound) over k of En. The set of variables ζnk is infinitesimal if and only if

lim Sn = 0 as n→∞.

(To be continued.)


HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins