A general strong law of large numbers for additive arithmetic functions

TL;DR

This paper establishes a broad strong law of large numbers for strongly additive arithmetic functions, showing convergence of weighted sums of i.i.d. variables under mild conditions.

Contribution

It proves a general weighted strong law of large numbers for strongly additive functions, extending classical results to more complex arithmetic functions.

Findings

Weighted sums of i.i.d. variables converge almost surely to the expected value.

The result applies under mild conditions on the additive function.

It generalizes classical laws of large numbers to a broader class of functions.

Abstract

Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables, then $$ \lim_{N\to \infty} {\sum_{n=1}^N f(n) X_n \over\sum_{n=1}^N f(n)} \buildrel{a.s.}\over{=} \E X . $$