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

Consider a sequence {xi} of independent chance vectors in k dimensions with identical distributions, and a sequence of mutually exclusive events S1, S2, ⋯, such that Sidepends only on the first i vectors and Σ P(Si) = 1. Let φibe a real or complex function of the first i vectors in the sequence sati...

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Bibliographic Details
Published in:The Annals of mathematical statistics 1946-09, Vol.17 (3), p.310-317
Main Authors: Blackwell, D., Girshick, M. A.
Format: Article
Language:English
Subjects:
Coefficients
Coordinate systems
Degrees of polynomials
Generating function
Mathematics
Polynomials
Probabilities
Random variables
Random walk
Randomness
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Summary:Consider a sequence {xi} of independent chance vectors in k dimensions with identical distributions, and a sequence of mutually exclusive events S1, S2, ⋯, such that Sidepends only on the first i vectors and Σ P(Si) = 1. Let φibe a real or complex function of the first i vectors in the sequence satisfying conditions: (1) E(φi) = O and (2) E(φj∣ X1, ⋯, Xi) = φifor j ≥ i. Let φ = φiand n = i when Sioccurs. A general theorem is proved which gives the conditions φimust satisfy such that Eφ = 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 φ 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.
ISSN:0003-4851
2168-8990
DOI:10.1214/aoms/1177730943