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Almost Sure Convergence of Renewal Processes

  • Valeriĭ V. Buldygin
  • Karl-Heinz Indlekofer
  • Oleg I. Klesov
  • Josef G. Steinebach
Chapter
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 91)

Abstract

Consider some renewal sequence, that is, a sequence of partial sums {Sn}n≥0 of independent identically distributed random variables {Xn}n≥1. Our aim in this chapter is to show that various functionals of partial sums and corresponding renewal processes are asymptotically equivalent if one considers them from the point of view of generalized renewal processes.

References

  1. 11.
    G. Alsmeyer, Erneuerungstheorie. Analyse stochastischer Regenerationsschemata, Teubner, Stuttgart, 1991.CrossRefGoogle Scholar
  2. 13.
    G. Alsmeyer, On the Markov renewal theorem, Stochastic Process. Appl. 50 (1994), no. 1, 37–56.MathSciNetCrossRefGoogle Scholar
  3. 24.
    V.S. Barbu and N. Limnios, Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications: Their use in Reliability and DNA Analysis, Springer-Verlag, New York, 2008.zbMATHGoogle Scholar
  4. 47.
    A.A. Borovkov, Probability Theory, Springer, London, 2013.CrossRefGoogle Scholar
  5. 93.
    D.R. Cox, Renewal Theory, Wiley, New York, 1962.zbMATHGoogle Scholar
  6. 94.
    D.R. Cox and P.A.W. Lewis, The Statistical Analysis of Series of Events, Wiley, New York, 1966.CrossRefGoogle Scholar
  7. 111.
    J.L. Doob, Renewal theory from the point of view of the theory of probability, Trans. Amer. Math. Soc. 63 (1948), no. 3, 422–438.MathSciNetCrossRefGoogle Scholar
  8. 124.
    I. Fazekas and O.I. Klesov, A general approach to the strong laws of large numbers, Teor. Veroyatnost. i Primenen. 45 (2000), no. 3, 568–583 (Ukrainian); English transl. in Theory Probab. Appl. 45 (2002), no. 3, 436–449.MathSciNetCrossRefGoogle Scholar
  9. 126.
    W. Feller, A limit theorem for random variables with infinite moments, Amer. J. Math. 68 (1946), no. 2, 257–262.MathSciNetCrossRefGoogle Scholar
  10. 129.
    W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed., Wiley, New York, 1971.zbMATHGoogle Scholar
  11. 138.
    A. Frolov, A. Martikainen, and J. Steinebach, Limit theorems for maxima of sums and renewal processes, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Veroyatn. i Stat. 278 (2001), 261–274, 314–315 (Russian); English transl. in J. Math. Sci. (N. Y.) 118 (2003), no. 6, 5658–5666.Google Scholar
  12. 141.
    J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, 1978.zbMATHGoogle Scholar
  13. 159.
    A. Gut, Stopped Random Walks. Limit Theorems and Applications, second edition, Springer, New York, 2009.CrossRefGoogle Scholar
  14. 160.
    A. Gut, O. Klesov, and J. Steinebach, Equivalences in strong limit theorems for renewal counting processes, Statist. Probab. Lett. 35 (1997), no. 4, 381–394.MathSciNetCrossRefGoogle Scholar
  15. 184.
    L. Horváth, Strong approximation of extended renewal processes, Ann. Probab. 12 (1984), no. 4, 1149–1166.MathSciNetCrossRefGoogle Scholar
  16. 199.
    S. Janson, Renewal theory for m-dependent variables, Ann. Probab. 11 (1983), no. 3, 558–568.MathSciNetCrossRefGoogle Scholar
  17. 219.
    K. Khorshidian, Strong law of large numbers for nonlinear semi-Markov reward processes, Asian J. Math. Stat. 3 (2010), no. 4, 310–315.MathSciNetGoogle Scholar
  18. 227.
    O. Klesov, Z. Rychlik, and J. Steinebach, Strong limit theorems for general renewal processes, Probab. Math. Statist. 21 (2001), no. 2, 329–349.MathSciNetzbMATHGoogle Scholar
  19. 229.
    O.I. Klesov and J. Steinebach, Asymptotic behavior of renewal processes defined by random walks with multidimensional time, Teor. Imovirnost. Matem. Statist. 56 (1997), 105–111 (Ukrainian); English transl. in Theory Probab. Math. Statist. 56 (1998), 107–113.Google Scholar
  20. 234.
    A. Kolmogoroff, Sur la loi forte des grands nombres, C. R. Acad. Sci. Paris 191 (1930), 910–912.zbMATHGoogle Scholar
  21. 238.
    I.N. Kovalenko, Queueing theory, Progress in Science: Probability theory. Mathematical statistics. Theoretical cybernetics, 5–109, Akad. Nauk SSSR Vsesojuz. Inst. Naucn. i Tehn. Informacii, Moscow, 1970. (Russian)Google Scholar
  22. 251.
    L. Lipsky, Queueing Theory: A Linear Algebraic Approach, Springer-Verlas, New York, 2008.zbMATHGoogle Scholar
  23. 260.
    J. Marcinkiewicz and A. Zygmund, Sur les fonctions indépendantes, Fund. Math. 29 (1937), 60–90.zbMATHGoogle Scholar
  24. 264.
    A.I. Martikainen and V.V. Petrov, On a Feller theorem, Teor. Veroytnost. i Primenen 25 (1980), no. 1, 194–197 (Russian); English transl. in Theory Probab. Appl. 25 (1980), no. 1, 191–193.Google Scholar
  25. 278.
    T. Mori, The strong law of large numbers when extreme terms are excluded from sums, Z. Wahrscheinlichkeitstheorie verw. Gebiete 36 (1976), no. 3, 189–194.MathSciNetCrossRefGoogle Scholar
  26. 279.
    É. Mourier, Lois des grands nombres et théorie ergodique, C. R. Acad. Sci. Paris 232 (1951), 923-925.MathSciNetzbMATHGoogle Scholar
  27. 298.
    R. Pyke, Markov renewal processes: definitions and preliminary properties, Ann. Math. Stat. 32 (1961), no. 4, 1231–1242.MathSciNetCrossRefGoogle Scholar
  28. 326.
    R. Serfozo, Travel times in queueing networks and network sojourns, Ann. Oper. Res. 48 (1994), no. 1–4, 3–29.MathSciNetzbMATHGoogle Scholar
  29. 327.
    R. Serfozo, Basics of Applied Stochastic Processes, Springer-Verlag, Berlin, 2009.CrossRefGoogle Scholar
  30. 328.
    B.A. Sevast’yanov, Renewal theory, Itogi Nauki Tehn.; Ser. Teor. Verojatn., mat. Statist., teor. Kibernet. 11 (1974), 99-128 (Russian); English transl. in J. Soviet. Math., 4 (1975), 281–302 (1976).Google Scholar
  31. 331.
    Q.-M. Shao, Maximal inequalities for partial sums of ρ-mixing sequences, Ann. Probab. 23 (1995), no. 2, 948–965.MathSciNetCrossRefGoogle Scholar
  32. 333.
    V.M. Shurenkov, Ergodic Theorems and Related Problems, “Naukova Dumka”, Kiev, 1981 (Russian); English transl. VSP, Utrecht, 1998.Google Scholar
  33. 334.
    V.M. Shurenkov, Ergodic Markov Processes, “Nauka”, Moscow, 1989. (Russian)Google Scholar
  34. 335.
    D. Siegmund, Sequential Analysis. Tests and Confidence Intervals, Springer–Verlag, New York, 1985.Google Scholar
  35. 337.
    W.L. Smith, Renewal theory and its ramifications, J. Roy. Stat. Soc. B20 (1958), no. 2, 243–302.MathSciNetzbMATHGoogle Scholar
  36. 366.
    M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis, CBMS–NFS Regional Conf. Ser. Appl. Math., vol. 39, SIAM, Philadelphia, 1982.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Valeriĭ V. Buldygin
    • 1
  • Karl-Heinz Indlekofer
    • 2
  • Oleg I. Klesov
    • 3
  • Josef G. Steinebach
    • 4
  1. 1.Department of Mathematical AnalysisNational Technical University of UkraineKyivUkraine
  2. 2.Department of MathematicsUniversity of PaderbornPaderbornGermany
  3. 3.Department of Mathematical Analysis and Probability TheoryNational Technical University of UkraineKyivUkraine
  4. 4.Mathematical InstituteUniversity of CologneCologneGermany

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