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Maximum of Random Walk

Chapter
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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

Random Walk Independent Random Variable Tail Asymptotics Negative Drift Subexponential Distribution 
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.

References

  1. [1]
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)zbMATHGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle Scholar
  7. [7]
    Athreya, K., Ney, P.: Branching Processes. Springer, Berlin (1972)zbMATHCrossRefGoogle Scholar
  8. [8]
    Bertoin, J., Doney, R.A.: On the local behaviour of ladder height distributions. J. Appl. Probab. 31, 816–821 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  10. [10]
    Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Berlin (1976)zbMATHCrossRefGoogle Scholar
  11. [11]
    Borovkov, A.A., Borovkov, K.A.: Asymptotic Analysis of Random Walks. Heavy-Tailed Distributions. Cambridge University Press, Cambridge (2008)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Chover, J., Ney, P., Wainger, S.: Degeneracy properties of subcritical branching processes. Ann. Probab. 1, 663–673 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Cline, D.: Convolutions of distributions with exponential and subexponential tails. J. Aust. Math. Soc. 43, 347–365 (1987)MathSciNetzbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Embrechts, P., Goldie, C.M.: On convolution tails. Stoch. Proc. Appl. 13, 263–278 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, 335–347 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New York (1971)zbMATHGoogle Scholar
  27. [27]
    Foss, S., Korshunov, D.: Lower limits and equivalences for convolution tails. Ann. Probab. 35, 366–383 (2007)MathSciNetzbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle Scholar
  31. [31]
    Kalashnikov, V., Tsitsiashvili, G.: Tails of waiting times and their bounds. Queueing Syst. 32, 257–283 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Klüppelberg, C.: Subexponential distributions and integrated tails. J. Appl. Probab. 25, 132–141 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Klüppelberg, C.: Subexponential distributions and characterization of related classes. Probab. Theor. Rel. Fields 82, 259–269 (1989)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    Pakes, A.G.: On the tails of waiting-time distributions. J. Appl. Probab. 12, 555–564 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    Pitman, E.J.G.: Subexponential distribution functions. J. Aust. Math. Soc. Ser. A 29, 337–347 (1980)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    Sgibnev, M.S.: Banach algebras of functions that have identical asymptotic behaviour at infinity. Siberian Math. J. 22, 179–187 (1981)MathSciNetzbMATHGoogle Scholar
  50. [50]
    Seneta, E.: Regularly Varying Functions, Springer, Berlin (1976)zbMATHCrossRefGoogle Scholar
  51. [51]
    Sornette, D.: Critical Phenomena in Natural Sciences, 2nd edn. Springer, Berlin (2004)zbMATHGoogle Scholar
  52. [52]
    Teugels, J.L.: The class of subexponential distributions. Ann. Probab. 3, 1000–1011 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    Veraverbeke, N.: Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  54. [54]
    Zachary, S.: A note on Veraverbeke’s theorem. Queueing Syst. 46, 9–14 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Actuarial MathematicsHeriot-Watt UniversityEdinburghUK
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia

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