A Combinatorial Lemma and its Application to Probability Theory

Abstract

To explain the idea behind the present paper the following fundamental principle is emphasized. Let X = (X ₁,…, X _n ) be an n-dimensional vector valued random variable, and let µ(x) =µ(x ₁…, x _n )be its probability measure (defined on euclidean n-space E _n ). Suppose that X has the property that µ(x) =µ(gx) for every element g of a group G of order h of transformations of E _n into itself. Let f(x) =f(x ₁…, x _n ) be a µ-integrable complex valued function on E _n Then the expected value of f(x) is

$$Ef\left( X \right) = \smallint f\left( x \right)d\mu \left( x \right) = \smallint \bar f\left( x \right)d\mu \left( x \right)$$

(1.1)

, where

$$\bar f\left( x \right) = \frac{1}{h}\sum\limits_{g \in G} f \left( {gx} \right)$$

(1.2)

.

The research of this author was supported in part by the united States Air Force under Contract No. AF18(600)-685 monitored by the Office of Scientific Research.