Abstract
Let $\{S_n\}$ be a random walk with underlying distribution function $F(x)$ and $\{\gamma_n\}$ be a sequence of constants such that $\gamma_n/n$ is nondecreasing. A universal integral test is given which determines the lower limit of $S_n/\gamma_n$ up to a constant scale for $\lim \sup \gamma_{2n}/\gamma_n < \infty$. The generalized LIL is obtained which contains the main result of Fristedt-Pruitt (1971). The rapidly growing random walks and the limit points of $\{S_n/\gamma_n\}$ are also studied.
Citation
Cun-Hui Zhang. "The Lower Limit of a Normalized Random Walk." Ann. Probab. 14 (2) 560 - 581, April, 1986. https://doi.org/10.1214/aop/1176992531
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