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A local limit theorem for random walk maxima with heavy tails


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  • Asmussen, Søren
  • Kalashnikov, Vladimir
  • Konstantinides, Dimitrios
  • Klüppelberg, Claudia
  • Tsitsiashvili, Gurami
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    For a random walk with negative mean and heavy-tailed increment distribution F, it is well known that under suitable subexponential assumptions, the distribution [pi] of the maximum has a tail [pi](x,[infinity]) which is asymptotically proportional to . We supplement here this by a local result showing that [pi](x,x+z] is asymptotically proportional to zF(x,[infinity]).

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 56 (2002)
    Issue (Month): 4 (February)
    Pages: 399-404

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    Handle: RePEc:eee:stapro:v:56:y:2002:i:4:p:399-404

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    Keywords: Integrated tail Ladder height Subexponential distribution;


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    1. Embrechts, P. & Veraverbeke, N., 1982. "Estimates for the probability of ruin with special emphasis on the possibility of large claims," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 55-72, January.
    2. Kalashnikov, Vladimir & Konstantinides, Dimitrios, 2000. "Ruin under interest force and subexponential claims: a simple treatment," Insurance: Mathematics and Economics, Elsevier, vol. 27(1), pages 145-149, August.
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    Cited by:
    1. Barbe, Ph. & McCormick, W.P. & Zhang, C., 2007. "Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1835-1847, December.
    2. Gao, Qingwu & Wang, Yuebao, 2009. "Ruin probability and local ruin probability in the random multi-delayed renewal risk model," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 588-596, March.
    3. Hansen, Niels Richard & Jensen, Anders Tolver, 2005. "The extremal behaviour over regenerative cycles for Markov additive processes with heavy tails," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 579-591, April.
    4. Wang, Yuebao & Yang, Yang & Wang, Kaiyong & Cheng, Dongya, 2007. "Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 256-266, March.
    5. Geluk, J.L. & Frenk, J.B.G., 2011. "Renewal theory for random variables with a heavy tailed distribution and finite variance," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 77-82, January.


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