Abstract.
Let be independent identically distributed random variables with regularly varying distribution tails:
where α≤ min (1,β), and L and L W are slowly varying functions as t→∞. Set S n =X 1 +⋯+X n , ¯S n = max 0≤ k ≤ n S k . We find the asymptotic behavior of P (S n > x)→0 and P (¯S n > x)→0 as x→∞, give a criterion for ¯S ∞ <∞ a.s. and, under broad conditions, prove that P (¯S ∞ > x)˜c V(x)/W(x).
In case when distribution tails of X j admit regularly varying majorants or minorants we find sharp estimates for the mentioned above probabilities under study.
We also establish a joint distributional representation for the global maximum ¯S ∞ and the time η when it was attained in the form of a compound Poisson random vector.
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Received: 4 June 2001 / Revised version: 10 September 2002 / Published online: 21 February 2003
Research supported by INTAS (grant 00265) and the Russian Foundation for Basic Research (grant 02-01-00902)
Mathematics Subject Classification (2000): 60F99, 60F10, 60G50
Key words or phrases: Attraction domain of a stable law – Maximum of sums of random variables – Criterion for the maximum of sums – Large deviations
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Borovkov, A. Large deviations probabilities for random walks in the absence of finite expectations of jumps. Probab Theory Relat Fields 125, 421–446 (2003). https://doi.org/10.1007/s00440-002-0243-1
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DOI: https://doi.org/10.1007/s00440-002-0243-1