1994 | OriginalPaper | Chapter
On Sequences of Sums of Independent Random Vectors
Author : Wassily Hoeffding
Published in: The Collected Works of Wassily Hoeffding
Publisher: Springer New York
Included in: Professional Book Archive
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This paper is concerned with certain properties of the sequence S1, S2,…of the sums Sn = X1 + … + Xn of independent, identically distributed, k-dimensional random vectors X1X1, …, where k ≧ 1. Attention is restricted to vectors Xn with integer-valued components. Let A1, A2, … be a sequence of k-dimensional measurable sets and let N denote the least n for which S1 ∈ A1. The values S0 = 0, S1, S2, … may be thought of as the successive positions of a moving particle which starts at the origin. The particle is absorbed when it enters set A1 at time n, and N is the time at which absorption occurs. Let M denote the number of times the particle is at the origin prior to absorption (the number of integers n, where 0 ≦n < N, for which S1 = 0). For the special case PXn = -1 = PXn = 1} = 1/2 it is found that (1.1)$$ E(M) = E\left( {\left| {S_N } \right|} \right) $$ whenenr E(N) < ∞. Thus the expectcd number of times the particle is at the origin prior to absorption equals its cxpected distance from the origin at the moment of absorption, for any time-dependeut absorption boundary such that the expected time of absorption is finite. Some restriction like E(N) < ∞ is essential. Indeed, if N is the least n ≧ 1 such that S n = 0, equation (1.1) would imply 1 = 0. In this case E(N) = ∞.