Celebratio Mathematica — Blackwell

Probability

Works connected to Meyer Abraham Girshick

Filter the Bibliography List

year	title	people

			clear

D. Blackwell and 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. Stat. 17 : 3 (September 1946), pp. 310–317. MR 0017898 Zbl 0060.29007 article

Consider a sequence \( \{x_i\} \) of independent chance vectors in \( k \) dimensions with identical distributions, and a sequence of mutually exclusive events \( S_1 \), \( S_2, \dots \), such that \( S_i \) depends only on the first \( i \) vectors and \( \sum P(S_i) = 1 \). Let \( \varphi_i \) be a real or complex function of the first \( i \) vectors in the sequence satisfying conditions:

\( E(\varphi_i) = 0 \) and
\( E(\varphi_j \mid X_1, \dots, X_i) = \varphi_i \) 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.