A note on Feller's strong law of large numbers
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 \textit{R. Feller}, Am. J. Math. 68, 257- 262 (1946; Zbl 0060.787); \textit{H. Kesten}, Ann. Math. Stat. 41, 1173-1205 (1970; Zbl 0233.60062) and \textit{K. B. Erickson}, Trans. Am. Math. Soc. 185, 371-381 (1973; Zbl 0304.60016).
scientific article; zbMATH DE number 3984240