Abstract: Let $\{X,X_n,n\geq1\}$ be a stationary stochastic sequence of
independent, or $\varphi$-mixing, or $\rho$-mixing positive random variables, or $\{X,X_n,n\geq1\}$ be a positive random variable sequence such that $\{X_n-EX,n\geq1\}$ is a stationary ergodic martingale differences, and set $S_n=\sum\limits^n_{j=1}X_j$ for $n\geq1 $. This paper proves certain law of the iterated logarithm for properly normalized products of the partial sums, $\prod\limits^n_{j=1}S_j/n!\mu^n$ when $EX=\mu>0$ and $0<{\rm Var}(X)<\infty$.

Key words: Product of sum, Laws of the iterated logarithm, Mixing sequence.