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

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

MathSciNet review:
876477

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**R. N. Bhattacharya and R. Ranga Rao,*Normal approximation and asymptotic expansions*, John Wiley & Sons, New York-London-Sydney, 1976. Wiley Series in Probability and Mathematical Statistics. MR**0436272****[2]**Kai Lai Chung and Harry Pollard,*An extension of renewal theory*, Proc. Amer. Math. Soc.**3**(1952), 303–309. MR**0048734**, https://doi.org/10.1090/S0002-9939-1952-0048734-5**[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**, https://doi.org/10.1090/S0002-9947-1948-0025098-8**[4]**Paul Embrechts, Makoto Maejima, and Edward Omey,*A renewal theorem of Blackwell type*, Ann. Probab.**12**(1984), no. 2, 561–570. MR**735853****[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**, https://doi.org/10.1090/S0002-9904-1949-09203-0**[6]**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**, https://doi.org/10.1017/S0305004100064057**[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), no. 3, 391–409. MR**721635**, https://doi.org/10.1007/BF00532230**[9]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, Oxford, at the Clarendon Press, 1954. 3rd ed. MR**0067125****[10]**Karl-Heinz Indlekofer,*A mean-value theorem for multiplicative functions*, Math. Z.**172**(1980), no. 3, 255–271. MR**581443**, https://doi.org/10.1007/BF01215089**[11]**Tatsuo Kawata,*A theorem of renewal type*, Kōdai Math. Sem. Rep.**13**(1961), 185–194. MR**0141175****[12]**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****[13]**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**, https://doi.org/10.1017/S0305004100061399**[14]**P. Ney and S. Wainger,*The renewal theorem for a random walk in two-dimensional time*, Studia Math.**44**(1972), 71–85. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. MR**0322978**, https://doi.org/10.4064/sm-44-1-71-85**[15]**V. V. Petrov,*Sums of independent random variables*, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. MR**0388499****[16]**Walter L. Smith,*Renewal theory and its ramifications*, J. Roy. Statist. Soc. Ser. B**20**(1958), 243–302. MR**0099090**

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

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

Additional Information

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