On the moment of crossing the one-sided nonlinear boundary by sums of independent random variables

We consider the distribution of the stopping time $\tau=\inf\{n\ge 1:S_n\ge f(n)\}$, where $S_n$ is a sum of $n$ independent identically distributed random variables with zero mean. It is shown that for nondecreasing boundaries $f(n)$ the asymptotics of $\mathbf P\{\tau>n\}$ as $n\to\infty$ coinsides (up to some constant) with the asymptotics of the same probability corresponding to $f(n)=\mathrm{const}$ if $\mathbf E(S_1^+)^2<\infty$. An analogous result is obtained in the case of nonincreasing boundaries under the assumption $\mathbf E\operatorname{exp}(\lambda S_1)<\infty$ for some $\lambda>0$. We obtain also the asymptotics of $\mathbf P\{\tau>n\}$ in the case when $\mathbf E\tau<\infty$ and the probability $\mathbf P\{S_1<-x\}$ is regularly varying as $x\to\infty$.