The lower limit of a normalized random walk (Q1082719)

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.