Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] R. N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, Wiley, New York, 1976. MR 0436272 (55:9219)
  • [2] K. L. Chung and H. Pollard, An extension of renewal theory, Proc. Amer. Math. Soc. 3 (1952), 303-309. MR 0048734 (14:61d)
  • [3] J. L. Doob, Renewal theory from the point of view of the theory of probability, Trans. Amer. Math. Soc. 63 (1948), 422-438. MR 0025098 (9:598d)
  • [4] P. Embrechts, M. Maejima and E. Omey, A renewal theorem of Blackwell type, Ann. Probab. 12 (1984), 561-570. MR 735853 (85d:60160)
  • [5] P. Erdös, W. Feller and H. Pollard, A property of power series with positive coefficients, Bull. Amer. Math. Soc. 55 (1949), 201-204. MR 0027867 (10:367d)
  • [6] J. Galambos and I. Kátai, A note on random walks in multidimensional time, Math. Proc. Cambridge Philos. Soc. 99 (1986), 163-170. MR 809511 (87a:60078)
  • [7] -, Some remarks on random walks in multidimensional time, Proc. 5th Pannonian Sympos. on Math. Statist. (Visegrád, Hungary, 1985), J. Mogyoródi et al., eds., Reidel, Dordrecht, 1986a.
  • [8] P. Greenwood, E. Omey and J. L. Teugels, Harmonic renewal measures, Z. Wahrsch. Verw. Gebiete 59 (1982), 391-409. MR 721635 (85e:60093)
  • [9] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, 1960. MR 0067125 (16:673c)
  • [10] K.-H. Indlekofer, A mean-value theorem for multiplicative functions, Math. Z. 172 (1980), 255-271. MR 581443 (82d:10068)
  • [11] T. Kawata, A theorem of renewal type, Kodai Math. Sem. Rep. 13 (1961), 185-194. MR 0141175 (25:4586)
  • [12] Yu. V. Linnik and A. I. Vinogradov, An estimate of the sum of the number of divisors in some intervals of an arithmetical progression, Uspehi Mat. Nauk 12 (76) (1957), 277-280. MR 0094312 (20:831)
  • [13] M. Maejima and T. Mori, Some renewal theorems for random walks in multidimensional time, Math. Proc. Cambridge Philos. Soc. 95 (1984), 149-154. MR 727089 (85f:60129)
  • [14] P. Ney and S. Wainger, The renewal theorem for a random walk in two-dimensional time, Studia Math. 46 (1972), 71-85. MR 0322978 (48:1336)
  • [15] V. V. Petrov, Sums of independent random variables, Springer-Verlag, Heidelberg, 1975. MR 0388499 (52:9335)
  • [16] W. L. Smith, Renewal theory and its ramifications, J. Roy Statist. Soc. (B) 20 (1958), 243-302. MR 0099090 (20:5534)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60K05, 60F05

Retrieve articles in all journals with MSC: 60K05, 60F05

Additional Information

Keywords: Random walk, multidimensional time, renewal theorem, weighted renewal function, inequality, asymptotic formula, divisor function
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society