Infinitesimal Random Variables

San José State University

applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

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, ... , z_n, ... 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

z_n = u_n1 + u_n2 + ... + u_{nk_n}.

The set of random variables {{u_n1, u_n2, u_{nk_n}} for n=1, 2, ...} is infinitesimal if for any ε > 0
lim_n->∞[ sup_{1≤k≤k_n} Prob{|u_nk|≥ε}] = 0
where sup stands for supremum, the least upper bound.
Let P_nk(u) be the cumulative probability distribution of the random variable u_nk. Then the following lemma holds:
Lemma: The set of random variables
{u_nk, k=1,..,k_n; n=1, 2, ..}
is infinitesimal if and only if
for any ε > 0

lim_n->∞[sup_{1≤k≤k_n} ∫{(u²/(1+u²)dP_nk(u)}] = 0

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

Lemma: Let {ζ_n, n=1, 2, ...} be a sequence of random variables such that each is the sum of independent random variables; i.e., ζ_n = ζ_n1 + ζ_n2 +...+ ζ_{nk_n}. Let F_nk be the cumulative distribution function of the variable ζ_n. Let E_nk be the expected value of of x²/(1+x²) with respect to the cumulative distribution function F_nk; i.e.,

Let S_n be the supremum (least upper bound) over k of E_n. The set of variables ζ_nk is infinitesimal if and only if
lim S_n = 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.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins

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

limn->∞[ sup1≤k≤kn Prob{|unk|≥ε}] = 0

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

lim Sn = 0 as n→∞.

z_n = u_n1 + u_n2 + ... + u_{nk_n}.

lim_n->∞[ sup_{1≤k≤k_n} Prob{|u_nk|≥ε}] = 0

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

lim_n->∞[sup_{1≤k≤k_n} ∫{(u²/(1+u²)dP_nk(u)}] = 0

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

lim S_n = 0 as n→∞.