Estimates for the distributions of the sums of subexponential random variables

Let {S_n}_n=1 be a random walk with independent identically distributed increments {?_i}_i=1. We study the ratios of the probabilities P(S_n > x)/P(?₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S_t > x) ~ Et P(?₁ > x) as x ? 8. Here t is a positive integer-valued random variable independent of {?_i} _{i = 1} The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.