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Large deviations probabilities for random walks in the absence of finite expectations of jumps
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  • Published: March 2003

Large deviations probabilities for random walks in the absence of finite expectations of jumps

  • A.A. Borovkov1 

Probability Theory and Related Fields volume 125, pages 421–446 (2003)Cite this article

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  • 7 Citations

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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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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Koptyug pr. 4, Novosibirsk, 630090, Russia. e-mail: borovkov@math.nsc.ru, , , , , , RU

    A.A. Borovkov

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  1. A.A. Borovkov
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Additional information

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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  • Issue Date: March 2003

  • DOI: https://doi.org/10.1007/s00440-002-0243-1

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Keywords

  • Asymptotic Behavior
  • Random Vector
  • Global Maximum
  • Distributional Representation
  • Sharp Estimate
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