Law of Large Numbers for Monotone Convolution

TL;DR

This paper proves a law of large numbers for monotone convolutions of probability measures with finite variances, extending classical results to non-identical distributions using martingale convergence.

Contribution

It introduces a law of large numbers for monotone convolutions of non-identical probability measures with finite variances, utilizing martingale convergence theorem.

Findings

Established a law of large numbers for monotone convolutions.

Extended classical results to non-identical distributions.

Applied martingale convergence theorem in this context.

Abstract

Using martingale convergence theorem, we prove a law of large numbers for monotone convolutions $\mu_{1}\triangleright\mu_{2}\triangleright\cdots\triangleright\mu_{n}$, where $\mu_{j}$'s are probability laws on $\mathbb{R}$ with finite variances but not required to be identical.