Large deviations for random walks with nonidentically distributed jumps having infinite variance

Let $\xi_1,\xi_2,\dots$ be independent random variables with distributions $F_1,F_2,\dots$ in a triangular scheme ($F_i$ may depend on some parameter),
$$\mathbf{E}\xi_i=0, \quad S_n=\sum_{i=1}^n\xi_i, \quad \overline{S}_n=\max_{k\leqslant n}S_k.$$
Assuming that some regularly varying functions majorize and minorize $F=\frac1n\sum_{i=1}^nF_i$, we find upper and lower bounds for the probabilities $\mathbf{P}(S_n>x)$ and $\mathbf{P}(\overline{S}_n>z)$. These bounds are precise enough to yield asymptotics. We also study the asymptotics of the probability that a trajectory $\{S_k\}$ crosses the remote boundary $\{g(k)\}$ i.e., the asymptotics of $\mathbf{P}\bigl(\max_{k\leqslant n}(S_k-g(k))>0\bigr)$. The case $n=\infty$ is not exclude. We also estimate excluded. Ewlso estimate the disribution of the crossing time.