On the discrepancy of powers of random variables

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