Abstract. For a sequence of real-valued i.i.d. mean $0$ random variables $\{X, X_{n}; n \geq1 \} $ with partial sums $S_{n} = \sum_{i=1}^n X_{i}$, $n \geq1$, conditions are provided for $\{X, X_{n}; n \geq1 \} $ to enjoy one-sided iterated logarithm type behavior of the form $0 < \limsup_{n \rightarrow\infty } S_{n}/\sqrt{nh(n)} < \infty $ almost surely where $h(\cdot )$ is a positive, nondecreasing function which is slowly varying at infinity. New results are obtained as special cases and some open problems are posed.

AMS Subject Classification (1991): 60F15, 60G50

Keyword(s): Sums of i.i.d. random variables, law of the iterated logarithm, one-sided iterated logarithm type behavior, almost surely, slowly varying function

Received January 4, 2007, and in revised form May 10, 2007. (Registered under 5971/2009.)