On local probabilities of the crossings of nonlinear boundaries by sums of independent random values. (Russian) Zbl 0705.60041

Ragimov, F. G.

Teor. Veroyatn. Primen. 35, No. 2, 373-376 (1990).

Let \(\xi_ 1,\xi_ 2,...,\xi_ n,..\). be i.i.d. r.v. with \[ \mu =E \xi_ 1>0,\quad S_ n=\xi_ 1+...+\xi_ n,\quad \tau_ a=\inf \{n\geq a:\;S_ n\geq f_ a(n)+x\}, \] and there exists a sequence \(B_ n>0\) such that \[ \lim_{n\to \infty}P\{S_ n-n\mu \leq xB_ n\}=G_{\alpha,\beta}(x), \] where \(G_{\alpha,\beta}(x)\) is a distribution function of a stable law with parameters \(\alpha\in (1,2]\), \(\beta\in [-1,1]\). Asymptotic expansions for \[ P\{\tau_ a=n,\quad S_ n\geq f_ a(n)+x\} \] for some class of functions \(f_ a(n)\), \(n=n(a)\to \infty\), \(a\to \infty\), are given.

Reviewer: N.Leonenko

MSC:

60G50	Sums of independent random variables; random walks
60F05	Central limit and other weak theorems

Keywords:

stable distribution; local theorem; nonlinear boundaries; Asymptotic expansions