Stone’s decomposition of the renewal measure via Banach-algebraic techniques
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- by M. S. Sgibnev PDF
- 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
Additional Information
- M. S. Sgibnev
- Affiliation: Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 90, 630090 Russia
- Email: sgibnev@math.nsc.ru
- Received by editor(s): August 25, 2000
- Received by editor(s) in revised form: February 19, 2001
- Published electronically: February 4, 2002
- Additional Notes: This research was supported by Grant 99–01–00504 of the Russian Foundation of Basic Research.
- Communicated by: Claudia M. Neuhauser
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2425-2430
- MSC (2000): Primary 60K05
- DOI: https://doi.org/10.1090/S0002-9939-02-06317-7
- MathSciNet review: 1897469