A renewal theorem for random walks in multidimensional time

Authors:
J. Galambos, K.-H. Indlekofer and I. Kátai

Journal:
Trans. Amer. Math. Soc. **300** (1987), 759-769

MSC:
Primary 60K05; Secondary 60F05

MathSciNet review:
876477

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Abstract: Let be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let . The asymptotic behavior of the weighted sum , with summation over , is investigated as . In the special case , the number of solutions of the equation in positive integers becomes the renewal function for a random walk in -dimensional time whose terms are distributed as . Under some assumptions on the magnitude of and of , (i) it is shown that is asymptotically distribution free as , (ii) the proper order of magnitude of is determined, and under some further restrictions on , (iii) a simple asymptotic formula is given for . From (i), the known asymptotic formula for with or 3 is deduced under the sole assumption of finite variance. The relaxation of previous moment assumptions requires a new inequality for the sum of the divisor function , which by itself is of interest.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0876477-2

Keywords:
Random walk,
multidimensional time,
renewal theorem,
weighted renewal function,
inequality,
asymptotic formula,
divisor function

Article copyright:
© Copyright 1987
American Mathematical Society