Products of random variables and the first digit phenomenon

We provide conditions on dependent and on non-stationary random variables Xn ensuring that the mantissa of the sequence of products ∏1nXk is almost surely distributed following Benford's law or converges in distribution to Benford's law. This is achieved through proving new generalizations of Lévy's and Robbins's results on distribution modulo 1 of sums of independent random variables.