Estimates for the distributions of the sums of subexponential random variables

Let $\{S_n\}_{n\geqslant1}$ be a random walk with independent identically distributed increments $\{\xi_i\}_{i\geqslant1}$. We study the ratios of the probabilities $\mathbf{P}(S_n>x)/\mathbf{P}(\xi_1>1)$ 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 $\mathbf{P}(S_{\tau}>x)\sim\mathbf{E}_{\tau}\mathbf{P}(\xi_1>x)$ as $x\to\infty$. Here $\tau$ is a positive integer-valued random variable independent of $\{\xi_i\}_{i\geqslant1}$. 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.