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On the asymptotic behavior of distributions of first-passage times. II. (English. Russian original) Zbl 1138.60035
Math. Notes 75, No. 3, 322-330 (2004); translation from Mat. Zametki 75, No. 3, 350-359 (2004).
The two-part paper deals with the asymptotic behavior of $$P(\eta_-(x)>n)$$ resp. $$P(\eta_+(x)=n),$$ $$\eta_+(x)=\min\{k:S_k>x\},$$ $$\eta_-(x)=\min\{k:S_k\leq-x\}, S_n=\sum_{j=1}^n\xi_j$$, $$\xi,\xi_1,\xi_2,\ldots$$ being i.i.d. random variables. While in part I [Math. Notes 75, No. 1, 23–37 (2004); translation from Mat. Zametki 75, No. 1, 24–39 (2004; Zbl 1108.60039)], $$x=0$$ and $$n\to\infty$$ for some classes of distributions of $$\xi$$ is considered, in part II here, $$x\geq0$$ or even $$x\to\infty$$ is studied. The asymptotic behavior depends on $$E\xi$$ and, whether the sums $$D_+=\sum_k{P(S_k>0)\over k},$$ $$D_-=\sum_k{P(S_k\leq0)\over k}$$ converge or diverge. A rich list of references is given (22 items).

##### MSC:
 60G50 Sums of independent random variables; random walks 62E20 Asymptotic distribution theory in statistics
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