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Asymptotic analysis for random walks with nonidentically distributed jumps having finite variance. (Russian, English) Zbl 1117.60024
Sib. Mat. Zh. 46, No. 6, 1265-1287 (2005); translation in Sib. Math. J. 46, No. 6, 1020-1038 (2005).
Summary: 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 $$S_n=\sum_{i=1}^{n}\xi_i$$, $$\overline{S}_n=\max_{k\leq n} S_k$$. Assuming further that some regularly varying functions majorize or minorize the averaged distribution $$F_n=\frac1n\sum_{i=1}^{n}F_i$$, 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 $$\mathbf{P} (\max_{k\leq n} (S_k-g(k))>0)$$. The case $$n = \infty$$ is not excluded. We also estimate the distribution of the first crossing time.
##### MSC:
 60F10 Large deviations 60G50 Sums of independent random variables; random walks
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