On Functions of Sequences of Independent Chance Vectors with Applications to the Problem of the "Random Walk" in $k$ Dimensions

Abstract

Consider a sequence $\{x_i\}$ of independent chance vectors in $k$ dimensions with identical distributions, and a sequence of mutually exclusive events $S_1, S_2, \cdots$, such that $S_i$ depends only on the first $i$ vectors and $\Sigma P(S_i) = 1$. Let $\varphi_i$ be a real or complex function of the first $i$ vectors in the sequence satisfying conditions: (1) $E(\varphi_i) = O$ and (2) $E(\varphi_j \mid X_1, \cdots, X_i) = \varphi_i$ for $j \geq i$. Let $\varphi = \varphi_i$ and $n = i$ when $S_i$ occurs. A general theorem is proved which gives the conditions $\varphi_i$ must satisfy such that $E\varphi = 0$. This theorem generalizes some of the important results obtained by Wald for $k = 1$. A method is also given for obtaining the distribution of $\varphi$ and $n$ in the problem of the "random walk" in $k$ dimensions for the case in which the components of the vector take on a finite number of integral values.