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September, 1946 On Functions of Sequences of Independent Chance Vectors with Applications to the Problem of the "Random Walk" in $k$ Dimensions
D. Blackwell, M. A. Girshick
Ann. Math. Statist. 17(3): 310-317 (September, 1946). DOI: 10.1214/aoms/1177730943

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.

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D. Blackwell. M. A. Girshick. "On Functions of Sequences of Independent Chance Vectors with Applications to the Problem of the "Random Walk" in $k$ Dimensions." Ann. Math. Statist. 17 (3) 310 - 317, September, 1946. https://doi.org/10.1214/aoms/1177730943

Information

Published: September, 1946
First available in Project Euclid: 28 April 2007

zbMATH: 0060.29007
MathSciNet: MR17898
Digital Object Identifier: 10.1214/aoms/1177730943

Rights: Copyright © 1946 Institute of Mathematical Statistics

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Vol.17 • No. 3 • September, 1946
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