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Maximum on a random time interval of a random walk with infinite mean

Author

Listed:
  • Denis Denisov

    (University of Manchester)

Abstract

Let $$\xi _1,\xi _2,\ldots $$ ξ 1 , ξ 2 , … be independent, identically distributed random variables with infinite mean $${\mathbf {E}}[|\xi _1|]=\infty .$$ E [ | ξ 1 | ] = ∞ . Consider a random walk $$S_n=\xi _1+\cdots +\xi _n$$ S n = ξ 1 + ⋯ + ξ n , a stopping time $$\tau =\min \{n\ge 1: S_n\le 0\}$$ τ = min { n ≥ 1 : S n ≤ 0 } and let $$M_\tau =\max _{0\le i\le \tau } S_i$$ M τ = max 0 ≤ i ≤ τ S i . We study the asymptotics for $${\mathbf {P}}(M_\tau >x),$$ P ( M τ > x ) , as $$x\rightarrow \infty $$ x → ∞ .

Suggested Citation

  • Denis Denisov, 2021. "Maximum on a random time interval of a random walk with infinite mean," Queueing Systems: Theory and Applications, Springer, vol. 98(3), pages 211-223, August.
    • Handle: RePEc:spr:queues:v:98:y:2021:i:3:d:10.1007_s11134-020-09661-z
    DOI: 10.1007/s11134-020-09661-z
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    References listed on IDEAS

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    1. Asmussen, Søren & Kalashnikov, Vladimir & Konstantinides, Dimitrios & Klüppelberg, Claudia & Tsitsiashvili, Gurami, 2002. "A local limit theorem for random walk maxima with heavy tails," Statistics & Probability Letters, Elsevier, vol. 56(4), pages 399-404, February.
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