On the discrepancy of powers of random variables

Let (dn) be a sequence of positive numbers and let (Xn) be a sequence of positive independent random variables. We provide an upper bound for the deviation between the distribution of the mantissaes of the first N terms of (Xndn) and the Benford's law. If dn goes to infinity at a rate at most polynomial, this deviation converges a.s. to 0 as N goes to infinity.