Stone’s decomposition of the renewal measure via Banach-algebraic techniques

Sgibnev, M.

doi:10.1090/S0002-9939-02-06317-7

Stone’s decomposition of the renewal measure via Banach-algebraic techniques
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Proc. Amer. Math. Soc. 130 (2002), 2425-2430 Request permission

Abstract:

A Banach-algebraic approach to Stone’s decomposition of the renewal measure is discussed. Estimates of the rate of convergence in a key renewal theorem are given.

References

Gerold Alsmeyer, Erneuerungstheorie, Teubner Skripten zur Mathematischen Stochastik. [Teubner Texts on Mathematical Stochastics], B. G. Teubner, Stuttgart, 1991 (German). Analyse stochastischer Regenerationsschemata. [Analysis of stochastic regeneration schemes]. MR 1119301, DOI 10.1007/978-3-663-09977-2
Elja Arjas, Esa Nummelin, and Richard L. Tweedie, Uniform limit theorems for non-singular renewal and Markov renewal processes, J. Appl. Probability 15 (1978), no. 1, 112–125. MR 467955, DOI 10.2307/3213241
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Rudolf Grübel, On subordinated distributions and generalized renewal measures, Ann. Probab. 15 (1987), no. 1, 394–415. MR 877612
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
B. A. Rogozin, Asymptotic analysis of the renewal function, Teor. Verojatnost. i Primenen. 21 (1976), no. 4, 689–706 (Russian, with English summary). MR 0420900
B. A. Rogozin and M. S. Sgibnev, Banach algebras of measures on the line, Sibirsk. Mat. Zh. 21 (1980), no. 2, 160–169, 239 (Russian). MR 569185
Manfred Schäl, Über Lösungen einer Erneuerungsgleichung, Abh. Math. Sem. Univ. Hamburg 36 (1971), 89–98 (German). MR 334340, DOI 10.1007/BF02995911
M. S. Sgibnev, Submultiplicative moments of the supremum of a random walk with negative drift, Statist. Probab. Lett. 32 (1997), no. 4, 377–383. MR 1602211, DOI 10.1016/S0167-7152(96)00097-1
Mikhail S. Sgibnev, Exact asymptotic behaviour in a renewal theorem for convolution equivalent distributions with exponential tails, SUT J. Math. 35 (1999), no. 2, 247–262. MR 1737881
M. S. Sgibnev, An asymptotic expansion for the distribution of the supremum of a random walk, Studia Math. 140 (2000), no. 1, 41–55. MR 1763881, DOI 10.4064/sm-140-1-41-55
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
W. L. Smith, Remarks on the paper ‘Regenerative stochastic processes’, Proc. Roy. Soc. London Ser. A 256 (1960), 496–501. MR 115223, DOI 10.1098/rspa.1960.0121
Charles Stone, On absolutely continuous components and renewal theory, Ann. Math. Statist. 37 (1966), 271–275. MR 196795, DOI 10.1214/aoms/1177699617
N. B. Engibaryan, Renewal equations on the half-line, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 1, 61–76 (Russian, with Russian summary); English transl., Izv. Math. 63 (1999), no. 1, 57–71. MR 1701838, DOI 10.1070/im1999v063n01ABEH000228