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Stone’s decomposition of the renewal measure via Banach-algebraic techniques. (English) Zbl 0996.60096
Let \(F\) be a probability distribution on \(\mathbb R\) with positive mean \(\mu\) and let \(H\) be the corresponding renewal measure. C. Stone [Ann. Math. Stat. 37, 271-275 (1966; Zbl 0147.16205)] showed that, if for some \(m\geq 1\) \(m\)-times convolution of \(F\) has a nonzero absolutely continuous component, then there exists a decomposition \(H=H_1+H_2\), where \(H_2\) is a finite measure and \(H_1\) is absolutely continuous with bounded continuous density \(h(x)\) such that \(\lim_{x\to+\infty}h(x)=\mu^{-1}\) and \(\lim_{x\to-\infty}h(x)=0\). A lot of estimations are based on the representation of \(H\) under some additional assumptions. Stone’s decomposition is proved by using Banach-algebraic techniques. The method allows to extract detailed information about the asymptotic properties of the terms \(H_1\) and \(H_2\). Under some additional restrictions of submultiplicative type, estimates of the rate of convergence in the key renewal theorem are obtained.

MSC:
60K05 Renewal theory
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[1] 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]. · Zbl 0727.60102
[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. · Zbl 0375.60095
[3] William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. · Zbl 0077.12201
[4] Rudolf Grübel, On subordinated distributions and generalized renewal measures, Ann. Probab. 15 (1987), no. 1, 394 – 415. · Zbl 0613.60007
[5] E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloquium Publications, vol. 31, Providence, RI, 1957. · Zbl 0078.10004
[6] B. A. Rogozin, Asymptotic analysis of the renewal function, Teor. Verojatnost. i Primenen. 21 (1976), no. 4, 689 – 706 (Russian, with English summary).
[7] 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). · Zbl 0457.46024
[8] Manfred Schäl, Über Lösungen einer Erneuerungsgleichung, Abh. Math. Sem. Univ. Hamburg 36 (1971), 89 – 98 (German). Collection of articles dedicated to Lothar Collatz on his sixtieth birthday. · Zbl 0218.60086 · doi:10.1007/BF02995911 · doi.org
[9] M. S. Sgibnev, Submultiplicative moments of the supremum of a random walk with negative drift, Statist. Probab. Lett. 32 (1997), no. 4, 377 – 383. · Zbl 0903.60055 · doi:10.1016/S0167-7152(96)00097-1 · doi.org
[10] 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. · Zbl 0958.60083
[11] M. S. Sgibnev, An asymptotic expansion for the distribution of the supremum of a random walk, Studia Math. 140 (2000), no. 1, 41 – 55. · Zbl 0962.60019
[12] W. L. Smith, Regenerative stochastic processes. Proc. R. Soc. London A 232 (1955), 6-31. · Zbl 0067.36301
[13] W. L. Smith, Remarks on the paper ’Regenerative stochastic processes’, Proc. Roy. Soc. London. Ser. A 256 (1960), 496 – 501. · Zbl 0119.14203 · doi:10.1098/rspa.1960.0121 · doi.org
[14] Charles Stone, On absolutely continuous components and renewal theory, Ann. Math. Statist. 37 (1966), 271 – 275. · Zbl 0147.16205 · doi:10.1214/aoms/1177699617 · doi.org
[15] 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. , https://doi.org/10.1070/im1999v063n01ABEH000228 Norair B. Yengibarian, Renewal equation on the whole line, Stochastic Process. Appl. 85 (2000), no. 2, 237 – 247. · Zbl 0997.60096 · doi:10.1016/S0304-4149(99)00076-9 · doi.org
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