Asymptotic analysis for random walks with nonidentically distributed jumps having finite variance

Let $\xi_1\xi_2,\dots$ be independent random variables with distributions $F_1,F_2,\dots$ in a triangular array scheme ($F_i$ may depend on some parameter). Assume that $\mathbf E\xi_i=0$, $\mathbf E\xi_i^2<\infty$ and put $S_n=\sum^n_{i=1}\xi_i$, $\overline S_n=\max_{k\leqslant n}S_k$. Assuming further that some regularly varying functions majorize or minorize the “averaged” distribution $F=\frac1n\sum^n_{i=1}F_i$, we find upper and lower bounds for the probabilities $\mathbf P(S_n>x)$ and $\mathbf P(\overline S_n>x)$. We also study the asymptotics of these probabilities and of the probabilities that a trajectory $\{S_k\}$ crosses the remote boundary $\{g(k)\}$; that is, the asymptotics of $\mathbf P(\max_{k\leqslant n}(S_k-g(k))>0)$. The case $n=\infty$ is not excluded. We also estimate the distribution of the first crossing time.