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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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