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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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