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


Author: M. S. Sgibnev
Journal: Proc. Amer. Math. Soc. 130 (2002), 2425-2430
MSC (2000): Primary 60K05
DOI: https://doi.org/10.1090/S0002-9939-02-06317-7
Published electronically: February 4, 2002
MathSciNet review: 1897469
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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 [Enhancements On Off] (What's this?)

  • 1. G. Alsmeyer, Erneuerungstheorie, B.G. Teubner, Stuttgart, 1991. MR 92f:60148
  • 2. E. Arjas, E. Nummelin, R. L. Tweedie, Uniform limit theorems for non-singular renewal and Markov renewal processes. J. Appl. Probab. 15 (1978), 112-125. MR 57:7798
  • 3. W. Feller, An Introduction to Probability Theory and Its Applications II, Wiley, New York, 1966. MR 35:1048
  • 4. R. Grübel, On subordinated distributions and generalized renewal measures. Ann. Probab. 15 (1987), 394-415. MR 88d:60045
  • 5. E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloquium Publications, vol. 31, Providence, RI, 1957. MR 19:664d
  • 6. B. A. Rogozin, Asymptotics of renewal functions. Theory Probab. Appl. 21 (1976), 669-686. MR 54:8911
  • 7. B. A. Rogozin, M. S. Sgibnev, Banach algebras of measures on the line. Siberian Math. J. 21 (1980), 265-273. MR 81e:43003
  • 8. M. Schäl, Über Lösungen einer Erneuerungsgleichung. Abh. Math. Sem. Univ. Hamburg 36 (1971), 89-98. MR 48:12659
  • 9. M. S. Sgibnev, Submultiplicative moments of the supremum of a random walk with negative drift. Statist. Probab. Lett. 32 (1997), 377-383. MR 99e:60158
  • 10. M. S. Sgibnev, Exact asymptotic behaviour in a renewal theorem for convolution equivalent distributions with exponential tails. SUT Journal of Mathematics 35 (1999), 247-262. MR 2000m:60102
  • 11. M. S. Sgibnev, An asymptotic expansion for the distribution of the supremum of a random walk. Studia Math. 140 (2000), 41-55. MR 2001g:60109
  • 12. W. L. Smith, Regenerative stochastic processes. Proc. R. Soc. London A 232 (1955), 6-31. MR 17:502b
  • 13. W. L. Smith, Remarks on the paper `Regenerative stochastic processes'. Proc. R. Soc. London A 256 (1960), 496-501. MR 22:6025
  • 14. C. Stone, On absolutely continuous components and renewal theory. Ann. Math. Statist. 37 (1966), 271-275. MR 33:4981
  • 15. N. B. Yengibarian, Renewal equation on the whole line. Stochastic Process. Appl. 85 (2000), 237-247.MR 2001d:60097b

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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

DOI: https://doi.org/10.1090/S0002-9939-02-06317-7
Keywords: Stone's decomposition, renewal measure, asymptotic behavior, submultiplicative function, spread-out distribution, Banach algebra
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
Article copyright: © Copyright 2002 American Mathematical Society

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