## A note on Feller’s strong law of large numbers.(English)Zbl 0608.60052

Let $$(X_ n)$$ be a sequence of i.i.d. random variables with $$S_ n=\sum^{n}_{j=1}X_ j$$, $$n\geq 1$$ and let $$(\gamma_ n)$$ be a sequence of positive constants such that $$\gamma_ n/n$$ is not decreasing in n. Define $$\gamma(x)=0$$ if $$x=0$$, $$=\gamma_ n$$ if $$x=n$$, $$n\geq 1$$ and $$=\gamma_ n+(\gamma_{n+1}-\gamma_ n)(x-n)$$ if $$n\leq x\leq n+1$$, $$n\geq 0$$. Let $$\gamma^{-1}(\cdot)$$ denote the inverse function of $$\gamma(\cdot)$$. Let $$x^+=\max(x,0)$$ and $$x^-=\max (- x,0)$$. Set $m_+(x)= \int^{x}_{0}P(X^+_ 1\geq t)dt,\quad m_- (x)= \int^{x}_{0}P(X^-_ 1\geq t)dt,$
$J_+(\gamma)= \int^{\infty}_{0} \min(\gamma^{-1}(x),x/m_+(x))dP(X_ 1\leq x)\quad and$
$J_-(\gamma)=\int^{\infty}_{0}\min (\gamma^{- 1}(x),x/m_-(x))dP(X_ 1>-x).$ The author shows that lim sup $$S_ n/\gamma_ n$$ is a.s. either $$\infty$$ or $$<\infty$$ (lim inf $$S_ n/\gamma_ n$$ is a.s. either $$-\infty$$ or $$>-\infty)$$ according as whether $$J_+(\gamma)=\infty$$ or $$<\infty$$ $$(J_-(\gamma)=\infty$$ or $$<\infty)$$. This is done by investigating the positive and negative contributions of $$S_ n$$ in relation to $$\gamma_ n$$. Two interesting corollaries have also been obtained.
This paper extends the results of R. Feller, Am. J. Math. 68, 257- 262 (1946; Zbl 0060.787); H. Kesten, Ann. Math. Stat. 41, 1173-1205 (1970; Zbl 0233.60062) and K. B. Erickson, Trans. Am. Math. Soc. 185, 371-381 (1973; Zbl 0304.60016).
Reviewer: R.Vasudeva

### MSC:

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

### Keywords:

integral tests; almost sure limit points

### Citations:

Zbl 0060.787; Zbl 0233.60062; Zbl 0304.60016
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