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

 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 , ˉS _n = max _{0≤ k ≤ n} S _k . We find the asymptotic behavior of P (S _n > x)→0 and P (ˉS _n > x)→0 as x→∞, give a criterion for ˉS _∞ <∞ a.s. and, under broad conditions, prove that P (ˉS _∞ > 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 ˉS _∞ and the time η when it was attained in the form of a compound Poisson random vector. 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