Extremes

, Volume 14, Issue 1, pp 63–125 | Cite as

Veraverbeke’s theorem at large: on the maximum of some processes with negative drift and heavy tail innovations

Article

Abstract

Veraverbeke’s (Stoch Proc Appl 5:27–37, 1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke’s theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems.

Keywords

Maximum of random walk Heavy tail Fractional ARIMA process Long range dependence Boundary crossing probability Nonlinear renewal theory 

AMS 2000 Subject Classifications

Primary—60G50; Secondary—60F99 60G99 60K30 62P05 62M10 26A12 26A33 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akonom, J., Gouriéroux, Chr.: A functional central limit theorem for fractional processes. Discussion paper 8801, CEPREMAP, Paris (1987)Google Scholar
  2. Asmussen, S.: Ruin Probabilities. World Scientific (2000)Google Scholar
  3. Asmussen, S., Klüppelberg, C.: Large deviation results for subexponential tails, with applications to insurance risk. Stoch. Process. their Appl. 64, 103–125 (1996)MATHCrossRefGoogle Scholar
  4. Billingsley, P.: Convergence of Probability Measures. Wiley (1968)Google Scholar
  5. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, 2nd edn. Cambridge University Press, Cambridge (1989)MATHGoogle Scholar
  6. Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Heidelberg (1971)Google Scholar
  7. Borovkov, A.A.: Large deviation probabilities for random walks in the absence of finite expectation jumps. Probab. Theory Relat. Fields 125, 421–446 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. Braaksma, B.L.J., Stark, D.: A Darboux-type theorem for slowly varying functions. J. Comb. Theory Ser. A 77, 51–66 (1997)MathSciNetMATHCrossRefGoogle Scholar
  9. Cline, D.B.H., Hsing, T.: Large deviation probability for sums of random variables with heavy or subexponential tails. Texas A&M Univ. (1991, preprint)Google Scholar
  10. Embrechts, P., Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insur. Math. Econ. 1, 55–77 (1982)MathSciNetMATHCrossRefGoogle Scholar
  11. Granger, C.W.J., Ding, Z.: Varieties of long memory models. J. Econom. 73, 61–77 (1996)MathSciNetMATHCrossRefGoogle Scholar
  12. Jamison, B., Orey, S., Pruitt, W.: Convergence of weighted averages of independent random variables. Zeit. Wahrsch. Theor. Verw. Geb. 4, 40–44 (1965)MathSciNetMATHCrossRefGoogle Scholar
  13. Konstantinides, D., Mikosch, T.: Large deviations and ruin probabilities for solutions of stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33, 1992–2035 (2005)MathSciNetMATHCrossRefGoogle Scholar
  14. Korshunov, D.: On distribution tail of the maximum of a random walk. Stoch. Process. Appl. 72, 97–103 (1997)MathSciNetMATHCrossRefGoogle Scholar
  15. Lindvall, T.: Weak convergence of probability measures and random functions in the function space D[ 0, ∞ ). J. Appl. Probab. 10, 109–121 (1973)MathSciNetMATHCrossRefGoogle Scholar
  16. Mikosch, T., Samorodnitsky, G.: The supremum of a negative drift random walk with dependent heavy-tail steps. Ann. Appl. Probab. 10, 1025–1064 (2000)MathSciNetMATHCrossRefGoogle Scholar
  17. Nagaev, A.: Integral limit theorem for large deviations when Cramér’s condition is not fulfilled, I, II. Theory Probab. Appl. 14, 51–64, 193–208 (1969a)MathSciNetCrossRefGoogle Scholar
  18. Nagaev, A.: Limit theorems for large deviations when Cramér’s conditions are violated. Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 6, 17–22 (in Russian) (1969b)MathSciNetMATHGoogle Scholar
  19. Ng, K.W., Tang, Q., Yan, J.-A., Yang, H.: Precise large deviations for sum of random variables with consistently varying tails. J. Appl. Probab. 41, 93–107 (2004)MathSciNetMATHCrossRefGoogle Scholar
  20. Pakes, A.G.: On the tails of waiting-time distributions. J. Appl. Probab. 12, 555–564 (1975)MathSciNetMATHCrossRefGoogle Scholar
  21. Pollard, D.: Convergence of Stochastic Processes. Springer, Heidelberg (1984)MATHGoogle Scholar
  22. Veraverbeke, N.: Asymptotic behavior of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37 (1977)MathSciNetMATHCrossRefGoogle Scholar
  23. Zachary, S.: A note on Veraverbeke’s theorem. Queueing Syst. 46, 9–14 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.CNRS (UMR 8088)ParisFrance
  2. 2.Dept. of StatisticsUniversity of GeorgiaAthensUSA

Personalised recommendations