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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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 $ X,\,{X_1},\,{X_2}, \ldots $ be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let $ {S_n} = {X_1} + \,{X_2} + \cdots + {X_n}$. The asymptotic behavior of the weighted sum $ R(k) = \sum {a_n}P({S_n} = k)$, with summation over $ n \geq 1$, is investigated as $ k \to + \infty $. In the special case $ {a_n} = {d_r}(n)$, the number of solutions of the equation $ n = {n_1}{n_2} \cdots {n_r}$ in positive integers $ {n_j},\,1 \leq j \leq r,\,R(k)$ becomes the renewal function $ Q(k)$ for a random walk in $ r$-dimensional time whose terms are distributed as $ X$. Under some assumptions on the magnitude of $ {a_n}$ and of $ A(x) = \sum\nolimits_{n \leq x} {{a_n}} $, (i) it is shown that $ R(k)$ is asymptotically distribution free as $ k \to + \infty $, (ii) the proper order of magnitude of $ R(k)$ is determined, and under some further restrictions on $ A(x)$, (iii) a simple asymptotic formula is given for $ R(k)$. From (i), the known asymptotic formula for $ Q(k)$ with $ r = 2$ 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 $ {d_r}(n),\,1 \leq n \leq x$, which by itself is of interest.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0876477-2
PII: S 0002-9947(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