# Functional limit theorems for random Lebesgue-Stieltjes convolutions

## TL;DR

This paper establishes joint functional limit theorems for Lebesgue-Stieltjes convolutions of nondecreasing stochastic processes, revealing similar asymptotic behaviors in standard and coupled branching random walks.

## Contribution

It provides the first joint functional limit theorems in the Skorokhod space for these convolutions, applicable to coupled branching random walks.

## Key findings

- Convolutions of nondecreasing processes converge in distribution in the Skorokhod space.
- Number of individuals in the $j$th generation shows similar asymptotic behavior.
- Results apply to both standard and coupled branching random walks.

## Abstract

We prove joint functional limit theorems in the Skorokhod space equipped with the $J_1$-topology for successive Lebesgue-Stieltjes convolutions of nondecreasing stochastic processes with themselves. These convolutions arise naturally in coupled branching random walks, where the displacements of individuals relative to their mother's position are given by the underlying point process rather than its copy. Surprisingly, the numbers of individuals in the $j$th generation, with positions less than or equal to $t$, exhibit remarkably similar distributional behavior in both standard branching random walks and coupled branching random walks as $t$ tends to infinity.