Maximum of Random Walk

  • Sergey Foss
  • Dmitry Korshunov
  • Stan Zachary
Chapter
Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)

Abstract

In this chapter, we study a random walk whose increments have a (right) heavy-tailed distribution with a negative mean. We also consider applications to queueing and risk processes.

Keywords

Income Summing 

References

  1. [1]
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)MATHGoogle Scholar
  2. [2]
    Asmussen, S.: Ruin Probabilities. World Scientific, Singapore (2000)Google Scholar
  3. [3]
    Asmussen, S., Foss, S.: On exceedance times for some processes with dependent increments. J. Appl. Probab. 51 (2014)Google Scholar
  4. [4]
    Asmussen, S., Foss, S., Korshunov, D.: Asymptotics for sums of random variables with local subexponential behaviour. J. Theor. Probab. 16, 489–518 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Asmussen, S., Kalashnikov, V., Konstantinides, D., Klüppelberg, C., Tsitsiashvili, G.: A local limit theorem for random walk maxima with heavy tails. Stat. Probab. Lett. 56, 399–404 (2002)MATHCrossRefGoogle Scholar
  6. [6]
    Asmussen, S., Klüppelberg, C.: Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Proc. Appl. 64, 103–125 (1996)MATHCrossRefGoogle Scholar
  7. [7]
    Athreya, K., Ney, P.: Branching Processes. Springer, Berlin (1972)MATHCrossRefGoogle Scholar
  8. [8]
    Bertoin, J., Doney, R.A.: On the local behaviour of ladder height distributions. J. Appl. Probab. 31, 816–821 (1994)MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  10. [10]
    Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Berlin (1976)MATHCrossRefGoogle Scholar
  11. [11]
    Borovkov, A.A., Borovkov, K.A.: Asymptotic Analysis of Random Walks. Heavy-Tailed Distributions. Cambridge University Press, Cambridge (2008)MATHCrossRefGoogle Scholar
  12. [12]
    Callaert, H., Cohen, J.W.: A lemma on regular variation of a transient renewal function. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 275–278 (1972)MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Chistyakov, V.P.: A theorem on sums of independent positive random variables and its application to branching random processes. Theor. Probab. Appl. 9, 640–648 (1964)CrossRefGoogle Scholar
  14. [14]
    Chover, J., Ney, P., Wainger, S.: Functions of probability measures. J. d’Analyse Mathématique 26, 255–302 (1973)MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Chover, J., Ney, P., Wainger, S.: Degeneracy properties of subcritical branching processes. Ann. Probab. 1, 663–673 (1973)MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Cline, D.: Convolutions of distributions with exponential and subexponential tails. J. Aust. Math. Soc. 43, 347–365 (1987)MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Cohen, J.W.: Some results on regular variation for distributions in queueing and fluctuation theory. J. Appl. Probab. 10, 343–353 (1973)MATHCrossRefGoogle Scholar
  18. [18]
    Denisov, D., Foss, S., Korshunov, D.: Tail asymptotics for the supremum of a random walk when the mean is not finite. Queueing Syst. 46, 15–33 (2004)MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Denisov, D., Foss, S., Korshunov, D.: On lower limits and equivalences for distribution tails of randomly stopped sums. Bernoulli 14, 391–404 (2008)MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Denisov, D., Foss, S., Korshunov, D.: Lower limits for distribution tails of randomly stopped sums. Theor. Probab. Appl. 52, 690–699 (2008)MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Embrechts, P., Goldie, C.M.: On closure and factorization theorems for subexponential and related distributions. J. Aust. Math. Soc. Ser. A 29, 243–256 (1980)MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Embrechts, P., Goldie, C.M.: On convolution tails. Stoch. Proc. Appl. 13, 263–278 (1982)MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, 335–347 (1979)MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)MATHCrossRefGoogle Scholar
  25. [25]
    Embrechts, P., Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insur. Math. Econ. 1, 55–72 (1982)MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New York (1971)MATHGoogle Scholar
  27. [27]
    Foss, S., Korshunov, D.: Lower limits and equivalences for convolution tails. Ann. Probab. 35, 366–383 (2007)MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Frisch, U., Sornette, D.: Extreme deviations and applications. J. Phys. I France 7, 1155–1171 (1997)CrossRefGoogle Scholar
  29. [29]
    Hallinan, A.J.: A review of the Weibull distribution. J. Qual. Tech. 25, 85–93 (1993)Google Scholar
  30. [30]
    Kalashnikov, V.: Geometric Sums: Bounds for Rare Events with Applications. Kluwer Academic Publishers, Dordrecht (1997)MATHCrossRefGoogle Scholar
  31. [31]
    Kalashnikov, V., Tsitsiashvili, G.: Tails of waiting times and their bounds. Queueing Syst. 32, 257–283 (1999)MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    Klüppelberg, C.: Subexponential distributions and integrated tails. J. Appl. Probab. 25, 132–141 (1988)MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Klüppelberg, C.: Subexponential distributions and characterization of related classes. Probab. Theor. Rel. Fields 82, 259–269 (1989)MATHCrossRefGoogle Scholar
  34. [34]
    Korolev, V.Yu., Bening, V.E., Shorgin, S.Ya.: Mathematical foundations of risk theory. Fizmatlit, Moscow (in Russian) (2007)Google Scholar
  35. [35]
    Korshunov, D.: On distribution tail of the maximum of a random walk. Stoch. Proc. Appl. 72, 97–103 (1997)MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    Korshunov, D.: Large-deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution. Theor. Probab. Appl. 46, 355–366 (2002)MathSciNetCrossRefGoogle Scholar
  37. [37]
    Korshunov, D.: On the distribution density of the supremum of a random walk in the subexponential case. Siberian Math. J. 47, 1060–1065 (2006)MathSciNetCrossRefGoogle Scholar
  38. [38]
    Korshunov, D.: How to measure the accuracy of the subexponential approximation for the stationary single server queue. Queueing Syst. 68, 261–266 (2011)MathSciNetCrossRefGoogle Scholar
  39. [39]
    Laherrère, J., Sornette, D.: Streched exponential distributions in nature and economy: “fat tails” with characteristic scales. Eur. Phys. J. B 2, 525–539 (1998)CrossRefGoogle Scholar
  40. [40]
    Leslie, J.R.: On the non-closure under convolution of the subexponential family. J. Appl. Probab. 26, 58–66 (1989)MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    Lindley, D. V.: The theory of queues with a single server. Proc. Cambridge Philos. Soc. 8, 277–289 (1952)MathSciNetCrossRefGoogle Scholar
  42. [42]
    Malevergne, Y., Sornette, D.: Extreme Financial Risks: From Dependence to Risk Management. Springer, Heidelberg (2006)Google Scholar
  43. [43]
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    Pakes, A.G.: On the tails of waiting-time distributions. J. Appl. Probab. 12, 555–564 (1975)MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    Pitman, E.J.G.: Subexponential distribution functions. J. Aust. Math. Soc. Ser. A 29, 337–347 (1980)MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    Rogozin, B.A.: On the constant in the definition of subexponential distributions. Theor. Probab. Appl. 44, 409–412 (2000)MathSciNetCrossRefGoogle Scholar
  47. [47]
    Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, Chichester (1998)Google Scholar
  48. [48]
    Rudin, W.: Limits of ratios of tails of measures. Ann. Probab. 1, 982–994 (1973)MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    Sgibnev, M.S.: Banach algebras of functions that have identical asymptotic behaviour at infinity. Siberian Math. J. 22, 179–187 (1981)MathSciNetMATHGoogle Scholar
  50. [50]
    Seneta, E.: Regularly Varying Functions, Springer, Berlin (1976)MATHCrossRefGoogle Scholar
  51. [51]
    Sornette, D.: Critical Phenomena in Natural Sciences, 2nd edn. Springer, Berlin (2004)MATHGoogle Scholar
  52. [52]
    Teugels, J.L.: The class of subexponential distributions. Ann. Probab. 3, 1000–1011 (1975)MathSciNetMATHCrossRefGoogle Scholar
  53. [53]
    Veraverbeke, N.: Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37 (1977)MathSciNetMATHCrossRefGoogle Scholar
  54. [54]
    Zachary, S.: A note on Veraverbeke’s theorem. Queueing Syst. 46, 9–14 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media New York 2013

Authors and Affiliations

  • Sergey Foss
    • 1
  • Dmitry Korshunov
    • 2
  • Stan Zachary
    • 1
  1. 1.Department of Actuarial MathematicsHeriot-Watt UniversityEdinburghUK
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia

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