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

Abstract

Let ε₁, ε₂,... be independent random variables with distributions F₁, F₂,... in a triangular scheme (F _i may depend on some parameter), Eε _i = 0, and put $S_n = \sum\nolimits_{i = 1}^n \xi _i ,\bar S_n = \max _{k \leqslant n} S_k$. Assuming that some regularly varying functions majorize and minorize $F = \frac{1}{n}\sum\nolimits_{i = 1}^n {F_i }$, we find upper and lower bounds for the probabilities P(S _n > x) and P(${\bar S_n }$ > x). 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 P(max_k≤n(S _k − g(k)) > 0). The case n = ∞ is not exclude. We also estimate excluded the distribution of the first crossing time.