Advertisement

Asymptotics for Renewal Processes Constructed from Multi-indexed Random Walks

  • 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

In this chapter, we study the asymptotic behavior of the renewal function and that of the renewal process constructed from a random walk over a restricted domain of multidimensional time. In doing so, we essentially apply the Open image in new window property studied in Sect.  3.3.1 (see Chap.  3).

References

  1. 1.
    A. Abay, Renewal theorems for random walks in multidimensional time, Mathematica Slovaca 49 (1999), no. 3, 371–380.MathSciNetzbMATHGoogle Scholar
  2. 12.
    G. Alsmeyer, Some relations between harmonic renewal measures and certain first passage times, Statistics & Probability Letters 12 (1991), 19–27.MathSciNetCrossRefGoogle Scholar
  3. 49.
    A.A. Borovkov and K.A. Borovkov, Analogues of the Blackwell theorem for weighted renewal functions, Sibirsk. Mat. Zh. 55 (2014), no. 4, 724–743; English transl. in Sib. Math. J. 55 (2014), no. 4, 589–605.Google Scholar
  4. 56.
    V.V. Buldygin, K.-H. Indlekofer, O.I. Klesov, and J.G. Steinebach, Asymptotics of renewal processes: some recent developments, Ann. Univ. Sci. Budapest, Sect. Comp. 28 (2008), 107–139.Google Scholar
  5. 120.
    P. Erdös, On a theorem of Hsu and Robbins, Ann. Math. Statist. 20 (1949), no. 2, 286–291.MathSciNetCrossRefGoogle Scholar
  6. 122.
    P. Erdös, W. Feller, and H. Pollard, A property of power series with positive coefficients, Bull. Amer. Math. Soc. 55 (1949), no. 2, 201–204.MathSciNetCrossRefGoogle Scholar
  7. 125.
    W. Feller, On the integral equation of renewal theory, Ann. Math. Statist. 12 (1941), no. 3, 243–267.MathSciNetCrossRefGoogle Scholar
  8. 139.
    J.P. Gabriel, Martingales with a countable filtering index set, Ann. Probab. 5 (1977), no. 6, 888–898.MathSciNetCrossRefGoogle Scholar
  9. 142.
    J. Galambos, K.-H. Indlekofer, and I. Kátai, A renewal theorem for random walks in multidimensional time, Trans. Amer. Math. Soc. 300 (1987), no. 2, 759–769.MathSciNetCrossRefGoogle Scholar
  10. 143.
    J. Galambos and I. Kátai, A note on random walks in multidimensional time, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 163–170.MathSciNetCrossRefGoogle Scholar
  11. 144.
    J. Galambos and I. Kátai, Some remarks on random walks in multidimensional time, in Proc. 5th Pannonian Sympos. Math. Statist. (Visegrád, Hungary, 1985), Reidel, Dordrecht, 1988, pp. 65–74.Google Scholar
  12. 147.
    A.O. Gel’fond and Yu.V. Linnik, Elementary Methods in the Analytic Theory of Numbers, Fizmatlit, Moscow, 1962; English transl. Pergamon Press, New York, 1966.Google Scholar
  13. 153.
    P. Greenwood, E. Omey, and J.L. Teugels, Harmonic renewal measures, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 3, 391–409.MathSciNetCrossRefGoogle Scholar
  14. 158.
    A. Gut, Strong laws for independent identically distributed random variables indexed by a Sector, Ann. Probab. 11 (1983), no. 3, 569–577.MathSciNetCrossRefGoogle Scholar
  15. 169.
    C. Hagwood, A renewal theorem in multidimensional time, Australian J. Statist. 31 (1989), no. 1, 130–137.MathSciNetCrossRefGoogle Scholar
  16. 176.
    D.R. Heath-Brown, Mean values of the zeta-function and divisor problems, in Recent Progress in Analytic Number Theory, Academic Press, London–New York, 1981, pp. 115–119.Google Scholar
  17. 185.
    P.L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci U.S.A. 33 (1947), no. 2, 25–31.MathSciNetCrossRefGoogle Scholar
  18. 191.
    K.-H. Indlekofer, I. Kátai, and O.I. Klesov, Renewal theorems for some weighted renewal functions, Ann. Univ. Sci. Budapest., Sect. Comp. 34 (2011), 179–194.Google Scholar
  19. 192.
    K.-H. Indlekofer and O.I. Klesov, Strong law of large numbers for multiple sums whose indices belong to a sector with function boundaries, Teor. Veroyatnost. i Primenen. 52 (2007), no. 4, 803–810; English transl. in Theory Probab. Appl. 52 (2008), no. 4, 711–719.MathSciNetCrossRefGoogle Scholar
  20. 194.
    K.-H. Indlekofer and O.I. Klesov, The asymptotic behavior of the renewal process con- structed from a random walk with a restricted multidimensional time domain, Ann. Univ. Sci. Budapest, Sect. Comp. 24 (2004), 209–221.Google Scholar
  21. 196.
    K.-H. Indlekofer and O.I. Klesov, Strong law of large numbers for multiple sums whose indices belong to a sector with function boundaries, Teor. Veroyatnost. i Primenen. 52 (2007), no. 4, 803–810 (Russian); English transl. in Theory Probab. Appl. 52 (2008), no. 4, 711–719.MathSciNetCrossRefGoogle Scholar
  22. 197.
    B.G. Ivanoff and E. Merzbach, What is a multi-parameter renewal process?, Stochastics 78 (2006), no. 6, 411–441.MathSciNetCrossRefGoogle Scholar
  23. 208.
    I. Kátai, Personal communication, 2009.Google Scholar
  24. 224.
    O.I. Klesov, A renewal theorem for a random walk with multidimensional time, Ukrain. Matem. Zh. 43 (1991), no. 9, 1161–1167 (Russian); English transl. in Ukrain. Math. J. 43 (1991), no. 9, 1089–1094.Google Scholar
  25. 225.
    O.I. Klesov, Limit Theorems for Multi-Indexed Sums of Random Variables, Springer, Berlin–Heidelberg–New York, 2014.CrossRefGoogle Scholar
  26. 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
  27. 230.
    O.I. Klesov and J. Steinebach, The asymptotic behavior of the renewal function constructed from a random walk in multidimensional time with restricted time domain, Ann. Univ. Sci. Budapest, Sect. Comp. 22 (2003), 181–192.Google Scholar
  28. 233.
    G. Kolesnik, On the estimation of multiple exponential sums, in Recent Progress in Analytic Number Theory, Academic Press, London-New York, 1981, pp. 231–246.Google Scholar
  29. 243.
    E. Krätzel, Lattice Points, Mathematics and its Applications (East European Series), 33, Kluwer Academic Publishers Group, Dordrecht, 1988.Google Scholar
  30. 253.
    A.J. Lotka, A contribution to the theory of self-renewing aggregates, with special reference to industrial replacement, Ann. Math. Statist. 10 (1939), no. 1, 1–25.CrossRefGoogle Scholar
  31. 254.
    M. Maejima and T. Mori, Some renewal theorems for random walks in multidimensional time, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 149–154.MathSciNetCrossRefGoogle Scholar
  32. 283.
    P. Ney and S. Wainger, The renewal theorem for a random walk in two-dimensional time, Studia Math. 44 (1972), 71–85.MathSciNetCrossRefGoogle Scholar
  33. 288.
    E. Omey and J.L. Teugels, Weighted renewal functions: a hierarchical approach, Adv. Appl. Prob. 34 (2002), no. 2, 394–415.MathSciNetCrossRefGoogle Scholar
  34. 337.
    W.L. Smith, Renewal theory and its ramifications, J. Roy. Stat. Soc. B20 (1958), no. 2, 243–302.MathSciNetzbMATHGoogle Scholar
  35. 338.
    R.T. Smythe, Strong laws of large numbers for r-dimensional arrays of random variables, Ann. Probab. 1 (1973), no. 1, 164–170.MathSciNetCrossRefGoogle Scholar
  36. 339.
    R.T. Smythe, Sums of independent random variables on partially ordered sets, Ann. Probab. 2 (1974), no. 5, 906–917.MathSciNetCrossRefGoogle Scholar
  37. 346.
    A.J. Stam, Some theorems on harmonic renewal measures, Stoch. Process. Appl. 39 (1991), no. 2, 277–285.MathSciNetCrossRefGoogle 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

Personalised recommendations