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Long Head-Runs and Long Match Patterns

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Advances in Finance and Stochastics

Summary

The survey presents recent developments in a specific area of Extreme Value Theory that deals with long head runs and related statistics. The topic has applications in insurance, finance, reliability and computational biology (see, for instance, [6; 13]). In contrast to existing surveys, we do not restrict ourselves to any particular method but provide an overview of different approaches.

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References

  1. Alsmeyer G. (1988) Second order approximation for certain stopped sums in extended renewal theory. — Adv. Appl. Probab., v. 20, No 2, 391–410.

    Article  MathSciNet  MATH  Google Scholar 

  2. Arratia R., Gordon L. and Waterman M.S. (1986) An extreme value theory for sequence matching. — Ann. Statist., v. 14, No 3, p. 971–993.

    Article  MathSciNet  MATH  Google Scholar 

  3. Arratia R., Gordon L. and Waterman M.S. (1990) The Erdös-Rényi law in distribution, for coin tossing and sequence matching. — Ann. Statist., v. 18, No 2, p. 539–570.

    Article  MathSciNet  MATH  Google Scholar 

  4. Arratia R., Goldstein L. and Gordon L. (1990) Two moments suffice for Poisson approximation. — Ann. Probab., v. 17, No 1, p. 9–25.

    Article  MathSciNet  Google Scholar 

  5. Arratia R. and Waterman M.S. (1989) The Erdös-Rényi strong law for pattern matching with a given proportion of mismatches. — Ann. Probab., v. 17, No 4, p. 1152–1169.

    Article  MathSciNet  MATH  Google Scholar 

  6. Barbour A.D. and Chryssaphinou O. (2001) Compoun Poisson approximation: a user’s guide. — University of Zürich: Preprint. http://www.math.unizh.ch/ adb/preprints.html

    Google Scholar 

  7. Barbour A.D., Holst L. and Janson S. (1992) Poisson Approximation. Oxford: Clarendon Press, 277 pp.

    Google Scholar 

  8. Berman S.M. (1962) Limiting distribution of the maximum term in sequences of dependent random variables. — Ann. Math. Statist., v. 33, 894–908.

    Article  MathSciNet  MATH  Google Scholar 

  9. Binswanger K. and Embrechts P. (1994) Longest runs in coin tossing. — Insurance: Math. Econ., v. 15, 139–149.

    Article  MathSciNet  MATH  Google Scholar 

  10. Deheuvels P., Devroye L. and Lynch J. (1986) Exact convergence rates in the limit theorems of Erdös-Rényi and Shepp. — Ann. Probab., v. 14, No 1, p. 209–223.

    Article  MathSciNet  MATH  Google Scholar 

  11. Deheuvels P. and Devroye L. (1987) Limit laws of Erdös-Rényi-Shepp type. — Ann. Probab., v. 15, No 4, p. 1363–1386.

    Article  MathSciNet  MATH  Google Scholar 

  12. Deheuvels P., Erdös P., Grill K. and Révész P. (1987) Many heads in a short block. — Math. Statist. Probab. Theory (M.L.Puri et al., eds.), v. A, p. 53–67.

    Google Scholar 

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

    Book  MATH  Google Scholar 

  14. Erdös P. and Rényi A. (1970) On a new law of large numbers. — J. Anal. Math., v. 22, p. 103–111.

    Article  Google Scholar 

  15. Frolov A.N. and Martikainen A.I. (1999) On the length of the longest increasing run in Rd. — Statist. Probab. Letters, v. 41, 153–161.

    Article  MathSciNet  MATH  Google Scholar 

  16. Geske M.X., Godbole A.P., Schaffner A.A., Skolnik A.M. and Wallstrom G.L. (1995) Compound Poisson approximation for word patterns under Markovian hypotheses. — J. Appl. Probab., v. 32, 877–892.

    Google Scholar 

  17. Goncharov V.L. (1944) On the field of combimatory analysis. — Amer. Math. Soc. Trani., v. 19, No 2, 1–46.

    MATH  Google Scholar 

  18. Grill K. (1987) Erdös-Révész type bounds for the length of the longest run from a stationary mixing sequence. — Probab. Theory Rel. Fields, v. 75, No 3, 77–85.

    Article  MathSciNet  MATH  Google Scholar 

  19. Karlin S. and Ost F. (1987) Counts of long aligned word matches among random letter sequences. — Adv. Appl. Probab., v. 19, No 2, 293–351.

    Article  MathSciNet  MATH  Google Scholar 

  20. Karlin S. and Ost F. (1988) Maximal length of common words among random letter sequences. — Ann. Probab., v. 16, No 3, 535–563.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kozlov A.M. and Piterbarg V.I. (2000) On large jumps of Cramer random walk. — Theory Probab. Appl. (submitted).

    Google Scholar 

  22. Mori T.F. More on the waiting time till each of some given patterns occurs as a run. — Can. J. Math., 1990, v. XLII, No 5, p. 915–932.

    Google Scholar 

  23. Mori T.F. (1991) On the waiting time till each of some given patterns occurs as a run. — Probab. Theory Rel. Fields, v. 87, 313–323.

    Article  MATH  Google Scholar 

  24. Novak S.Y. (1988) Time intervals of constant sojourn of a homogeneous Markov chain in a fixed subset of states. — Siberian Math. J., v. 29, No 1, 100–109.

    MathSciNet  MATH  Google Scholar 

  25. Novak S.Y. (1989) Asymptotic expansions in the problem of the longest head-run for a Markov chain with two states. — Trudy Inst. Math. (Novosibirsk), 1989, v. 13, p. 136–147 (in Russian).

    Google Scholar 

  26. Novak S.Y. (1992) Longest runs in a sequence of m-dependent random variables. — Probab. Theory Rel. Fields, v. 91, 269–281.

    Article  MATH  Google Scholar 

  27. Novak S.Y. (1993) On the asymptotic distribution of the number of random variables exceeding a given level. — Siberian Adv. Math., v. 3, No 4, 108–122.

    MathSciNet  Google Scholar 

  28. Novak S.Y. (1994) Asymptotic expansions for the maximum of random number of random variables. — Stochastic Process. Appl., v. 51, No 2, 297–305.

    MathSciNet  MATH  Google Scholar 

  29. Novak S.Y. (1994) Poisson approximation for the number of long “repetitions” in random sequences. — Theory Probab. Appl., v. 39, No 4, 593–603.

    MathSciNet  Google Scholar 

  30. Novak S.Y. (1995) Long match patterns in random sequences. — Siberian Adv. Math., v. 5, No 3, 128–140.

    MathSciNet  MATH  Google Scholar 

  31. Novak S.Y. (1997) On the Erdös-Rényi maximum of partial sums. — Theory Probab. Appl., v. 42, No 3, 254–270.

    Google Scholar 

  32. Novak S.Y. (1998) On the limiting distribition of extremes. — Siberian Adv. Math., v. 8, No 2, 70–95.

    MathSciNet  Google Scholar 

  33. Piterbarg V.I. (1991) On big jumps of a random walk. — Theory Probab. Appl., v. 36, No 1, p. 50–62.

    MathSciNet  MATH  Google Scholar 

  34. Révész P. (1990) Random walk in random and non-random environments Singapore: World Scientific, 332 p..

    Google Scholar 

  35. Reinert G. and Schbath S. (1998) Compound Poisson and Poisson process approximations for occurrences of multiple words. — J. Comput. Biol. 5, 223–253.

    Article  Google Scholar 

  36. Samarova S.S. (1981) On the length of the longest head-run for the Markov chain with two states. — Theory Probab. Appl., v. 26, No 3, 499–509.

    MathSciNet  Google Scholar 

  37. Smith R.L. Extreme value theory for dependent sequences via the Stein-Chen method of Poisson approximation. — Stoch. Proc. Appl., 1988, v. 30, No 2, p. 317–327.

    Article  MATH  Google Scholar 

  38. Zubkov A.M. and Mihailov V.G. (1979) On the repetitions of s-tuples in a sequence of independent trials. — Theory Probab. Appl., v. 24, No 2, p. 267–279.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Departement of Mathematics, Eidgenössische Technische Hochschule, Zurich, Switzerland

    Paul Embrechts

  2. Department of Mathematical Sciences, Brunel University, Uxbridge, UK

    Sergei Y. Novak

Authors
  1. Paul Embrechts
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  2. Sergei Y. Novak
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Editor information

Editors and Affiliations

  1. Lehrstuhl für Bankbetriebslehre, Johannes Gutenberg-Universität Mainz, Jakob Welder-Weg 9, 55128, Mainz, Germany

    Klaus Sandmann

  2. Inst. f. Gesellschafts- u. Wirtschaftswissenschaften Statistische Abteilung, Rheinische Friedrich-Wilhelms-Universität Bonn, Adenauerallee 24-42, 53113, Bonn, Germany

    Philipp J. Schönbucher

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© 2002 Springer-Verlag Berlin Heidelberg

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Embrechts, P., Novak, S.Y. (2002). Long Head-Runs and Long Match Patterns. In: Sandmann, K., Schönbucher, P.J. (eds) Advances in Finance and Stochastics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04790-3_3

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  • DOI: https://doi.org/10.1007/978-3-662-04790-3_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07792-0

  • Online ISBN: 978-3-662-04790-3

  • eBook Packages: Springer Book Archive

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