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

Let xi(1), xi(2),... be independent random variables with distributions F-1, F-2,... in a triangular array scheme (F-i may depend on some parameter). Assume that E xi(i) = 0, E xi(i)(2) < infinity, and put S-n = Sigma(i=1)(n) xi(i), (S) over bar (n) = max(k <= n) S-k. Assuming further that some regularly varying functions majorize or minorize the "averaged" distribution F = 1/n Sigma(i=1)(n) F-i, we find upper and lower bounds for the probabilities P(S-n > x) and P((S) over bar (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 P (max(k <= n)(S-k-g(k)) > 0). The case n = infinity is not excluded. We also estimate the distribution of the first crossing time.