Estimates for the distributions of the sums of subexponential random variables

Let {S-n}(ngreater than or equal to1) be a random walk with independent identically distributed increments {xi(i)}(igreater than or equal to1). We study the ratios of the probabilities P(S-n > x)/P( xi(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-tau > x) similar to EtauP( xi(1) > x) as x --> infinity. Here tau is a positive integer-valued random variable independent of {xi(i)}(igreater than or equal to1). 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.