A renewal theorem for random walks in multidimensional time

Galambos, J.; Indlekofer, K.-H.; Kátai, I.

doi:10.1090/S0002-9947-1987-0876477-2

A renewal theorem for random walks in multidimensional time
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by J. Galambos, K.-H. Indlekofer and I. Kátai PDF

Trans. Amer. Math. Soc. 300 (1987), 759-769 Request permission

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.

References

R. N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1976. MR 0436272
Kai Lai Chung and Harry Pollard, An extension of renewal theory, Proc. Amer. Math. Soc. 3 (1952), 303–309. MR 48734, DOI 10.1090/S0002-9939-1952-0048734-5
J. L. Doob, Renewal theory from the point of view of the theory of probability, Trans. Amer. Math. Soc. 63 (1948), 422–438. MR 25098, DOI 10.1090/S0002-9947-1948-0025098-8
Paul Embrechts, Makoto Maejima, and Edward Omey, A renewal theorem of Blackwell type, Ann. Probab. 12 (1984), no. 2, 561–570. MR 735853
P. Erdös, W. Feller, and H. Pollard, A property of power series with positive coefficients, Bull. Amer. Math. Soc. 55 (1949), 201–204. MR 27867, DOI 10.1090/S0002-9904-1949-09203-0
Janos Galambos and Imre Kátai, A note on random walks in multidimensional time, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 163–170. MR 809511, DOI 10.1017/S0305004100064057

Some remarks on random walks in multidimensional time

P. Greenwood, E. Omey, and J. L. Teugels, Harmonic renewal measures, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 3, 391–409. MR 721635, DOI 10.1007/BF00532230
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
Karl-Heinz Indlekofer, A mean-value theorem for multiplicative functions, Math. Z. 172 (1980), no. 3, 255–271. MR 581443, DOI 10.1007/BF01215089
Tatsuo Kawata, A theorem of renewal type, K\B{o}dai Math. Sem. Rep. 13 (1961), 185–194. MR 141175
A. I. Vinogradov and Yu. V. Linnik, Estimate of the sum of the number of divisors in a short segment of an arithmetic progression, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 4(76), 277–280 (Russian). MR 0094312
Makoto Maejima and Toshio Mori, Some renewal theorems for random walks in multidimensional time, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 149–154. MR 727089, DOI 10.1017/S0305004100061399
P. Ney and S. Wainger, The renewal theorem for a random walk in two-dimensional time, Studia Math. 44 (1972), 71–85. MR 322978, DOI 10.4064/sm-44-1-71-85
V. V. Petrov, Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown. MR 0388499
Walter L. Smith, Renewal theory and its ramifications, J. Roy. Statist. Soc. Ser. B 20 (1958), 243–302. MR 99090