On Sequences of Sums of Independent Random Vectors

Abstract

This paper is concerned with certain properties of the sequence S ₁, S ₂,…of the sums S _n = X ₁ + … + X _n of independent, identically distributed, k-dimensional random vectors X ₁ X ₁, …, where k ≧ 1. Attention is restricted to vectors X _n with integer-valued components. Let A ₁, A ₂, … be a sequence of k-dimensional measurable sets and let N denote the least n for which S ₁ ∈ A ₁. The values S ₀ = 0, S ₁, S ₂, … 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 A ₁ 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 S ₁ = 0). For the special case P X _n = -1 = P X _n = 1} = 1/2 it is found that

$$ E(M) = E\left( {\left| {S_N } \right|} \right) $$

((1.1))

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) = ∞.