Renewal Theory for Random Walks with Positive Drift

  • Allan Gut
Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)


Throughout this chapter (Ω, \(\mathfrak{F}\), P) is a probability space on which everything is defined, {S n , n ≥ 0} is a random walk with positive drift, that is, S o = 0, \(S_n=\sum\nolimits^n_{n-1}X_k, \,\, n \geq 1\), where {X k k ≥ 1} is a sequence of i.i.d. random variables, and \(S_n \xrightarrow{{\textrm{a.s.}}} + \infty \,\, {\textrm{as}}\,\, n \rightarrow \infty \). We assume throughout this chapter, unless stated otherwise, that 0 < EX 1 = μ < ∞ (recall Theorems 2.8.2 and 2.8.3). In this section, however, no such assumption is necessary.


Random Walk Central Limit Theorem Passage Time Renewal Process Renewal Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

  1. 1.
    Aleškevičiene, A. (1975): On the local limit theorem for the first passage time across a barrier (in Russian). Litovsk. Mat. Sb. XV, 23-66.Google Scholar
  2. 36.
    Berbee, H.C.P. (1979): Random Walks with Stationary Increments and Renewal Theory. Mathematical Centre Tracts 112, Amsterdam.MATHGoogle Scholar
  3. 42.
    Bingham, N.H. and Goldie, CM. (1982): Probabilistic and deterministic averaging. Trans. Amer. Math. Soc. 269, 453-480.MATHCrossRefMathSciNetGoogle Scholar
  4. 45.
    Blackwell, D. (1948): A renewal theorem. Duke Math. J. 15, 145-150.MATHCrossRefMathSciNetGoogle Scholar
  5. 46.
    Blackwell, D. (1953): Extension of a renewal theorem. Pacific J. Math. 3, 315-320.MATHMathSciNetGoogle Scholar
  6. 54.
    Carlsson, H. (1983): Remainder term estimates of the renewal function. Ann. Probab. 11, 143-157.MATHCrossRefMathSciNetGoogle Scholar
  7. 55.
    Carlsson, H. and Nerman, O. (1986): An alternative proof of Lorden’s renewal inequality. Adv. in Appl. Probab. 18, 1015-1016.MATHCrossRefMathSciNetGoogle Scholar
  8. 62.
    Chow, Y.S. and Hsiung, A. (1976): Limiting behaviour of maxjn Sjj and the first passage times in a random walk with positive drift. Bull. Inst. Math. Acad. Sinica 4, 35-44.MATHMathSciNetGoogle Scholar
  9. 67.
    Chow, Y.S. and Robbins, H. (1963): A renewal theorem for random variables which are dependent or nonidentically distributed. Ann. Math. Statist. 34, 390-395.MATHCrossRefMathSciNetGoogle Scholar
  10. 68.
    Chow, Y.S., Robbins, H. and Siegmund, D. (1971): Great Expectations: The Theory of Optimal Stopping. Houghton-Miffiin, Boston, MA.MATHGoogle Scholar
  11. 70.
    Chow, Y.S. and Teicher, H. (1966): On second moments of stopping rules. Ann. Math. Statist. 37, 388-392.MATHCrossRefMathSciNetGoogle Scholar
  12. 73.
    Chung, K.L. (1974): A Course in Probability Theory, 2nd ed. Academic Press, New York.MATHGoogle Scholar
  13. 76.
    Chung, K.L. and Pollard, H. (1952): An extension of renewal theory. Proc. Amer. Math. Soc. 3, 303-309.MATHCrossRefMathSciNetGoogle Scholar
  14. 77.
    Chung, K.L. and Wolfowitz, J. (1952): On a limit theorem in renewal theory. Ann. of Math. (2) 55, 1-6.CrossRefMathSciNetGoogle Scholar
  15. 81.
    Cox, D.R, and Smith, W.L. (1953): A direct proof of a fundamental theorem of renewal theory. Skand. Aktuarietidskr. 36, 139-150.MathSciNetGoogle Scholar
  16. 86.
    Daley, D.J. (1980): Tight bounds for the renewal function of a random walk. Ann. Probab. 8, 615-621.MATHCrossRefMathSciNetGoogle Scholar
  17. 97.
    Doob, J.L. (1948): Renewal theory from the point of view of the theory of probability. Trans. Amer. Math. Soc. 63, 422-438.MATHCrossRefMathSciNetGoogle Scholar
  18. 109.
    Essén, M. (1973): Banach algebra methods in renewal theory. J. Analyse Math. 26, 303-336.MATHCrossRefMathSciNetGoogle Scholar
  19. 115.
    Feller, W. (1971): An Introduction of Probability Theory and Its Applications, Vol. 2, 2nd ed. John Wiley, New York.Google Scholar
  20. 116.
    Feller, W. and Orey, S. (1961): A renewal theorem. J. Math. Mech. 10, 619-624.MATHMathSciNetGoogle Scholar
  21. 128.
    Grübel, R. (1986): On harmonic renewal measures. Probab. Th. Rel. Fields 71, 393-404.MATHCrossRefGoogle Scholar
  22. 131.
    Gut, A. (1974a): On the moments and limit distributions of some first passage times. Ann. Probab. 2, 277-308.MATHCrossRefMathSciNetGoogle Scholar
  23. 133.
    Gut, A. (1974c): On convergence in r-mean of some first passage times and randomly indexed partial sums. Ann. Probab. 2, 321-323.MATHCrossRefMathSciNetGoogle Scholar
  24. 132.
    Gut, A. (1974b): On the moments of some first passage times for sums of dependent random variables. Stoch. Process. Appl. 2, 115-126.MATHCrossRefMathSciNetGoogle Scholar
  25. 135.
    Gut, A. (1975b): On a.s. and r-mean convergence of random processes with an application to first passage times. Z. Wahrsch. verw. Gebiete 31, 333-341.MATHCrossRefMathSciNetGoogle Scholar
  26. 136.
    Gut, A. (1983a): Renewal theory and ladder variables. In: Probability and Mathematical Statistics. Essays in honour of Carl-Gustav Esseen (Eds. A. Gut and L. Hoist), 25-39. Uppsala.Google Scholar
  27. 137.
    Gut, A. (1983b): Complete convergence and convergence rates for randomly indexed partial sums with an application to some first passage times. Acta Math. Acad. Sci. Hungar. 42, 225-232; Correction, ibid. 45 (1985), 235-236.Google Scholar
  28. 138.
    Gut, A. (1985): On the law of the iterated logarithm for randomly indexed partial sums with two applications. Studia Sci. Math. Hungar. 20, 63-69.MATHMathSciNetGoogle Scholar
  29. 145.
    Gut, A. (2007): Probability: A Graduate Course, Corr. 2nd printing. Springer-Verlag, New York.Google Scholar
  30. 148.
    Gut, A. and Janson, S. (1983): The limiting behaviour of certain stopped sums and some applications. Scand. J. Statist. 10, 281-292.MATHMathSciNetGoogle Scholar
  31. 158.
    Heyde, C.C. (1964): Two probability theorems and their applications to some first passage problems. J. Austral. Math. Soc. 4, 214-222.MATHCrossRefMathSciNetGoogle Scholar
  32. 159.
    Heyde, C.C. (1966): Some renewal theorems with applications to a first passage problem. Ann. Math. Statist. 37, 699-710.MATHCrossRefMathSciNetGoogle Scholar
  33. 160.
    Heyde, C.C. (1967a): Asymptotic renewal results for a natural generalization of classical renewal theory. J. Roy. Statist. Soc. Ser. B 29, 141-150.MATHMathSciNetGoogle Scholar
  34. 161.
    Heyde, C.C. (1967b): A limit theorem for random walks with drift. J. Appl. Probab. 4, 144-150.MATHCrossRefMathSciNetGoogle Scholar
  35. 178.
    Janson, S. (1983): Renewal theory for m-dependent variables. Ann. Probab. 11, 558-568.MATHCrossRefMathSciNetGoogle Scholar
  36. 179.
    Janson, S. (1986): Moments for first passage and last exit times, the minimum and related quantities for random walks with positive drift. Adv. in Appl. Probab. 18, 865-879MATHCrossRefMathSciNetGoogle Scholar
  37. 187.
    Kingman, J.F.C. (1968): The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30, 499-510.MATHMathSciNetGoogle Scholar
  38. 192.
    Lai, T.L. (1975): On uniform integrability in renewal theory. Bull Inst. Math. Acad. Sinica 3, 99-105.MATHMathSciNetGoogle Scholar
  39. 193.
    Lai, T.L. (1976): Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Probab. 4, 51-66.MATHCrossRefGoogle Scholar
  40. 197.
    Lai, T.L. and Siegmund, D. (1979): A nonlinear renewal theory with applications to sequential analysis II. Ann. Statist. 7, 60-76.MATHCrossRefMathSciNetGoogle Scholar
  41. 200.
    Lalley, S.P. (1984a): Limit theorems for first-passage times in linear and nonlinear renewal theory. Adv. in Appl. Probab. 16, 766-803.MATHCrossRefMathSciNetGoogle Scholar
  42. 217.
    Lorden, G. (1970): On excess over the boundary. Ann. Math. Statist. 41, 520-527.MATHCrossRefMathSciNetGoogle Scholar
  43. 218.
    Maejima, M. (1975): On local limit theorems and Blackwell’s renewal theorem for indépendent random variables. Ann. Inst. Statist. Math. 27, 507-520.MATHCrossRefMathSciNetGoogle Scholar
  44. 219.
    Maejima, M. and Mori, T. (1984): Some renewal theorems for random walks in multidimensional time. Math. Proc. Cambridge Philos. Soc. 95, 149-154.MATHCrossRefMathSciNetGoogle Scholar
  45. 230.
    Ney, P. and Wainger, S. (1972): The renewal theorem for a random walk in two-dimensional time. Studia Math. XLIV, 71-85.MathSciNetGoogle Scholar
  46. 237.
    Prabhu, N.U. (1965): Stochastic Processes. Macmillan, New York.Google Scholar
  47. 238.
    Prabhu, N.U. (1980): Stochastic Storage Processes. Queues, Insurance Risk, and Dams. Springer-Verlag, New York.MATHGoogle Scholar
  48. 241.
    Rényi, A. (1957): On the asymptotic distribution of the sum of a random number of independent random variables. Ada Math. Acad. Sci. Hungar. 8, 193-199.MATHCrossRefGoogle Scholar
  49. 253.
    Siegmund, D.O. (1969): The variance of one-sided stopping rules. Ann. Math. Statist. 40, 1074-1077.MATHCrossRefMathSciNetGoogle Scholar
  50. 255.
    Siegmund, D. (1985): Sequential Analysis. Tests and Confidence Intervals. Springer-Verlag, New York.MATHGoogle Scholar
  51. 260.
    Smith, W.L. (1964): On the elementary renewal theorem for nonidentically distributed variables. Pacific J. Math. 14, 673-699.MATHMathSciNetGoogle Scholar
  52. 261.
    Smith, W.L. (1967): A theorem on functions of characteristic functions and its applications to some renewal theoretic random walk problems. In: Proc. Fifth Berkeley Symp. Math. Statist, and Probability, Vol. II/II, 265-309. University of California Press, Berkeley, CA.Google Scholar
  53. 267.
    Spitzer, F. (1960): A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 150-169.MATHCrossRefMathSciNetGoogle Scholar
  54. 269.
    Spitzer, F. (1976): Principles of Random Walk, 2nd ed. Springer-Verlag, New York.MATHGoogle Scholar
  55. 287.
    Stone, C. (1965): On characteristic functions and renewal theory. Trans. Amer. Math. Soc. 120, 327-342.MATHCrossRefMathSciNetGoogle Scholar
  56. 289.
    Stone, C. and Wainger, S. (1967): One-sided error estimates in renewal theory. J. Analyse Math. 20, 325-352.MATHCrossRefMathSciNetGoogle Scholar
  57. 298.
    Täcklind, S. (1944): Elementare Behandlung vom Erneuerungsproblem für den stationären Fall. Skand. Aktuarietidskr. 27, 1-15.MathSciNetGoogle Scholar
  58. 315.
    Williamson, J.A. (1965): Some renewal theorems for non-negative independent random variables. Trans. Amer. Math. Soc. 114, 417-445.MATHCrossRefMathSciNetGoogle Scholar
  59. 317.
    Woodroofe, M. (1976): A renewal theorem for curved boundaries and moments of first passage times. Ann. Probab. 4, 67-80.MATHCrossRefMathSciNetGoogle Scholar
  60. 319.
    Woodroofe, M. (1982): Nonlinear Renewal Theory in Sequential Analysis. CBMS-NSF Regional Gonf. Ser. in Appl. Math. 39. SIAM, Philadelphia, PA.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations