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), 759769
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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 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), 303309. 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), 422438. MR 0025098 (9:598d)
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 P. Embrechts, M. Maejima and E. Omey, A renewal theorem of Blackwell type, Ann. Probab. 12 (1984), 561570. MR 735853 (85d:60160)
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 P. Erdös, W. Feller and H. Pollard, A property of power series with positive coefficients, Bull. Amer. Math. Soc. 55 (1949), 201204. MR 0027867 (10:367d)
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 [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.
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 P. Greenwood, E. Omey and J. L. Teugels, Harmonic renewal measures, Z. Wahrsch. Verw. Gebiete 59 (1982), 391409. MR 721635 (85e:60093)
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 T. Kawata, A theorem of renewal type, Kodai Math. Sem. Rep. 13 (1961), 185194. MR 0141175 (25:4586)
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 M. Maejima and T. Mori, Some renewal theorems for random walks in multidimensional time, Math. Proc. Cambridge Philos. Soc. 95 (1984), 149154. MR 727089 (85f:60129)
 [14]
 P. Ney and S. Wainger, The renewal theorem for a random walk in twodimensional time, Studia Math. 46 (1972), 7185. MR 0322978 (48:1336)
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 V. V. Petrov, Sums of independent random variables, SpringerVerlag, Heidelberg, 1975. MR 0388499 (52:9335)
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 W. L. Smith, Renewal theory and its ramifications, J. Roy Statist. Soc. (B) 20 (1958), 243302. MR 0099090 (20:5534)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708764772
PII:
S 00029947(1987)08764772
Keywords:
Random walk,
multidimensional time,
renewal theorem,
weighted renewal function,
inequality,
asymptotic formula,
divisor function
Article copyright:
© Copyright 1987
American Mathematical Society
