# On the discrepancy of powers of random variables

## TL;DR

This paper investigates how the distribution of the mantissas of powered independent random variables converges to Benford's law, providing bounds and conditions for almost sure convergence.

## Contribution

It offers an upper bound on the deviation from Benford's law for powers of random variables and establishes almost sure convergence under polynomial growth of exponents.

## Key findings

- Deviation converges to zero almost surely for polynomial growth of exponents.
- Provides explicit upper bounds for the deviation from Benford's law.
- Demonstrates convergence behavior of mantissa distributions of powered variables.

## Abstract

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.