Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Precise large deviations for dependent regularly varying sequences
Download PDF
Download PDF
  • Published: 31 July 2012

Precise large deviations for dependent regularly varying sequences

  • Thomas Mikosch1 &
  • Olivier Wintenberger2 

Probability Theory and Related Fields volume 156, pages 851–887 (2013)Cite this article

  • 452 Accesses

  • 23 Citations

  • Metrics details

Abstract

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64, 193–208, 1969) and Nagaev (Ann Probab 7:745–789, 1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (Stoch Proc Appl 44:291–327, 1993; 68:1–20, 1997) in the context of central limit theorems with infinite variance stable limits. We illustrate the principle for stochastic volatility models, real valued functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Alsmeyer G.: On the Harris recurrence and iterated random Lipschitz functions and related convergence rate results. J. Theor. Probab. 16, 217–247 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andersen, T.G., Davis, R.A., Kreiss, J.-P., Mikosch, T. (eds.): The Handbook of Financial Time Series. Springer, Heidelberg (2009)

  3. Bartkiewicz K., Jakubowski A., Mikosch T., Wintenberger O.: Stable limits for sums of dependent infinite variance random variables. Probab. Theory Relat. Fields 150, 337–372 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Basrak B., Davis R.A., Mikosch T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908–920 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Basrak B., Davis R.A., Mikosch T.: Regular variation of GARCH processes. Stoch. Proc. Appl. 99, 95–116 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Basrak B., Segers J.: Regularly varying multivariate time series. Stoch. Proc. Appl. 119, 1055–1080 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Basrak, B., Segers, J.: A functional limit theorem for dependent sequences with infinite variance stable limits. Ann. Probab. (2011, to appear)

  8. Bertail P., Clémencon S.: Sharp bounds for the tails of functionals of Harris Markov chains. Theory Probab. Appl. 54, 505–515 (2009)

    Article  Google Scholar 

  9. Bingham N.H., Goldie C.M., Teugels J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  10. Breiman L.: On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10, 323–331 (1965)

    Article  MathSciNet  Google Scholar 

  11. Buraczewski, D., Damek, E., Mikosch, T., Zienkiewicz, J.: Large deviations for solutions to stochastic recurrence equations under Kesten’s condition. Ann. Probab. (2011, to appear)

  12. Cline, D.B.H., Hsing, T.: Large deviation probabilities for sums of random variables with heavy or subexponential tails, Technical Report, Texas A& M University (1998)

  13. Davis R.A., Hsing T.: Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879–917 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Davis R.A., Mikosch T.: Point process convergence of stochastic volatility processes with application to sample autocorrelation. J. Appl. Probab. 38A, 93–104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Davis R.A., Mikosch T.: Extremes of stochastic volatility models. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P., Mikosch, T. (eds.) Handbook of Financial Time Series, pp. 355–364. Springer, Berlin (2009)

    Chapter  Google Scholar 

  16. Dembo A., Zeitouni O.: Large Deviations Techniques and Applications. Corrected reprint of the second (1998) edition. Springer, Berlin (2010)

    Book  Google Scholar 

  17. Denisov D., Dieker A.B., Shneer V.: Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36, 1946–1991 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Doukhan, P.: Mixing. Properties and Examples. In: Lecture Notes in Statistics 85. Springer, New York (1994)

  19. Embrechts P., Klüppelberg C., Mikosch T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gantert N.: A not on logarithmic tail asymptotics and mixing. Stat. Probab. Lett. 49, 113–118 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Goldie C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hult H., Lindskog F.: Extremal behavior of regularly varying stochastic processes. Stoch. Proc. Appl. 115, 249–274 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hult H., Lindskog F.: Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94), 121–140 (2006)

    Article  MathSciNet  Google Scholar 

  25. Hult H., Lindskog F., Mikosch T., Samorodnitsky G.: Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Probab. 15, 2651–2680 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ibragimov I.A.: On the spectrum of stationary Gaussian sequences which satisfy the strong mixing condition II. Sufficient conditions. The rate of mixing. Theory Probab. Appl. 15, 24–37 (1970)

    MATH  Google Scholar 

  27. Jakubowski A.: Minimal conditions in p-stable limit theorems. Stoch. Proc. Appl. 44, 291–327 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jakubowski A.: Minimal conditions in p-stable limit theorems - II. Stoch. Proc. Appl. 68, 1–20 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jessen A.H., Mikosch T.: Regularly varying functions. Publ. Inst. Math. Nouvelle Série 80(94), 171–192 (2006)

    Article  MathSciNet  Google Scholar 

  30. Kallenberg O.: Random Measures, 3rd edn. Akademie-Verlag, Berlin (1983)

    MATH  Google Scholar 

  31. Kesten H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kolmogorov A.N., Rozanov Yu., A.: On the strong mixing conditions for stationary Gaussian sequences. Theory Probab. Appl. 5, 204–207 (1960)

    Article  MathSciNet  Google Scholar 

  33. Konstantinides D., Mikosch T.: Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33, 1992–2035 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Leadbetter M.R., Lindgren G., Rootzén H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  35. Leadbetter M.R., Rootzén H.: Extremal theory for stochastic processes. Ann. Probab. 16, 431–478 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Letac, G.: A contraction principle for certain Markov chains and its applications. In: Cohen, J.E., Kesten, H., Newman, C.M. (eds.) Random Matrices and Their Applications. Contemp. Math., vol. 50, pp. 263–273 (1986)

  37. Lesigne E., Volný D.: Large deviations for martingales. Stoch. Proc. Appl. 96, 143–159 (2001)

    Article  MATH  Google Scholar 

  38. Meyn S.P., Tweedie R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)

    Book  MATH  Google Scholar 

  39. Mikosch T., Samorodnitsky G.: The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10, 1025–1064 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mirek, M.: Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Probab. Relat. Fields (2011, to appear)

  41. Mogulskii, A.A.: Integral and integro-local theorems for sums of random variables with semi-exponential distribution (in Russian). Sib. Electr. Math. Rep. 6, 251–271 (2009)

  42. Nagaev, A.V.: Integral limit theorems for large deviations when Cramér’s condition is not fulfilled I, II. Theory Probab. Appl. 14, 51–64 and 193–208 (1969)

    Google Scholar 

  43. Nagaev S.V.: Large deviations of sums of independent random variables. Ann. Probab. 7, 745–789 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  44. Nummelin E.: General Irreducible Markov Chains and Non-Negative Operators. Cambridge University Press, Cambridge (1984)

    Book  MATH  Google Scholar 

  45. Petrov V.V.: Limit Theorems of Probability Theory. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  46. Pham T.D., Tran L.T.: Some mixing properties of time series models. Stoch. Proc. Appl. 19, 297–303 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  47. Pitman J.: Occupation measures for Markov chains. Adv. Appl. Probab. 9, 69–86 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  48. Resnick S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)

    MATH  Google Scholar 

  49. Resnick S.I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007)

    Google Scholar 

  50. Rio E.: Théorie asymptotique des processus aléatoires faiblement dépendants. Springer, Berlin (2000)

    MATH  Google Scholar 

  51. Samur J.D.: A regularity condition and a limit theorem for Harris ergodic Markov chains. Stoch. Proc. Appl. 111, 207–235 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Thomas Mikosch

  2. Centre De Recherche en Mathématiques de la Décision UMR CNRS 7534, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    Olivier Wintenberger

Authors
  1. Thomas Mikosch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olivier Wintenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olivier Wintenberger.

Additional information

T. Mikosch’s research is partly supported by the Danish Research Council (FNU) Grants 272-06-0442 and 09-072331. The research of T. Mikosch and O. Wintenberger is partly supported by a Danish-French Scientific Collaboration Grant of the French Embassy in Denmark. Both authors would like to thank their home institutions for hospitality when visiting each other.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mikosch, T., Wintenberger, O. Precise large deviations for dependent regularly varying sequences. Probab. Theory Relat. Fields 156, 851–887 (2013). https://doi.org/10.1007/s00440-012-0445-0

Download citation

  • Received: 15 November 2011

  • Accepted: 10 July 2012

  • Published: 31 July 2012

  • Issue Date: August 2013

  • DOI: https://doi.org/10.1007/s00440-012-0445-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stationary sequence
  • Large deviation principle
  • Regular variation
  • Markov processes
  • Stochastic volatility model
  • GARCH

Mathematics Subject Classification (2000)

  • Primary 60F10
  • Secondary 60J05
  • 60G70
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature