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
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) = ∞.
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Hoeffding, W. (1994). On Sequences of Sums of Independent Random Vectors. In: Fisher, N.I., Sen, P.K. (eds) The Collected Works of Wassily Hoeffding. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0865-5_25
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DOI: https://doi.org/10.1007/978-1-4612-0865-5_25
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