Frank Spitzer

A Combinatorial Lemma and its Application to Probability Theory

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

A limit theorem related to a new class of self similar processes

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)

Electrostatic Capacity, Heat Flow, and Brownian Motion

(1.1) , where

Convergence in Distribution of Products of Random Matrices

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

Recurrent Random Walk and Logarithmic Potential

(1.2) .

The characterization of optimal saving programs in a quadratic model

For a gentle approach to the problems connected with interacting Markov processes we review first what is known in the absence of interaction. For simplicity let S be a countable (or finite) set, and consider a countable (or finite if S is finite) collection of independent Markov processes on S,with common transition function P t (x,y), x, y ∈ S. It is natural to assume constant invariant measure, i.e.,