On the asymptotic behavior of distributions of first-passage times. II. (English. Russian original) Zbl 1138.60035

Borovkov, A. A.

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).

Reviewer: Peter Neumann (Dresden)

MSC:

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

Keywords:

random walk; first-passage time; asymptotics of the distribution of first-passage times