On the asymptotic behavior of distributions of first-passage times. II
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).
asymptotics of the distribution of first-passage times
scientific article; zbMATH DE number 2121361