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