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

Denis Denisov ()
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Denis Denisov: University of Manchester

Queueing Systems: Theory and Applications, 2021, vol. 98, issue 3, No 2, 223 pages

Abstract: 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 → ∞ .

Keywords: Random walk; Subexponential distribution; Heavy-tailed distribution; Busy period; Busy cycle; Single-server queue; Primary 60G70; Secondary 60K30; 60K25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s11134-020-09661-z

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