The lower limit of a normalized random walk.(English)Zbl 0603.60065

Let $$S_ n=X_ 1+...+X_ n$$, $$n\geq 1$$, where $$\{X_ n,n\geq 1\}$$ is a sequence of independent non-negative random variables with common distribution function F(x). The paper gives an integral test for the determination of lim inf $$S_ n/\gamma_ n$$, where $$\{\gamma_ n,n\geq 1\}$$ is a sequence of positive constants such that $$\gamma_ n/n$$ is non-decreasing in n. To formulate the main results, let $$\gamma$$ (x)/x be a continuous non-decreasing function such that $$\gamma (n)=\gamma_ n$$, $$n\geq 1$$, $$\beta^{-1}(x)$$ be the left-continuous inverse of $$\gamma$$ (x)/x and $I(\lambda)=\int^{\infty}_{1}x^{-1}\exp [-\beta^{- 1}(m(x)/\lambda)m(x)/x]dx,\quad \lambda >0,$ where $$m(x)=\int^{x}_{0}(1-F(t))dt$$. It is shown that
(i) if lim sup $$\gamma$$ $${}_{2n}/\gamma_ n=\rho <\infty$$, then $$\rho^{-1}\max (1.17,4\rho^{-1})\theta \leq \liminf S_ n/\gamma_ n\leq \rho e\theta$$, where $$\theta =\inf \{\lambda:I(\lambda)=\infty \};$$
(ii) if m(x) is a slowly varying function and inf$$\{\limsup_{n<k<wn} \gamma_ k/\gamma_ n:$$ $$w>1\}=1$$, then lim inf $$S_ n/\gamma_ n=\theta$$ a.s.;
(iii) if lim $$S_ n=\infty$$ a.s., then either $$P\{| X_ i| >x\}$$ is a slowly varying function as $$x\to \infty$$ or there exist constants $$b_ n$$, $$n\geq 1$$, such that $$0<b_ n\to \infty$$, $$b_ n/n$$ is non-decreasing and lim inf $$S_ n/b_ n=1$$ a.s.
Reviewer: F.Papangelou

MSC:

 60G50 Sums of independent random variables; random walks 60F15 Strong limit theorems 60F20 Zero-one laws
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