Exponential rate of L_p-convergence of intrinsic martingales in supercritical branching random walks

TL;DR

This paper establishes criteria for the exponential convergence rate of intrinsic martingales in supercritical branching random walks within the Lp space, enhancing understanding of their convergence behavior.

Contribution

It provides new criteria for the Lp-convergence rate of intrinsic martingales in supercritical branching random walks, a novel analysis in this context.

Findings

Criteria for Lp-convergence of the series involving intrinsic martingales.

Quantitative description of the exponential convergence rate.

Insights into the behavior of moments of the martingale differences.

Abstract

Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with almost sure limit $W$ in a supercritical branching random walk. We provide criteria for the $L_p$-convergence of the series $\sum_{n\ge 0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a statement about the exponential rate of convergence of $\me |W-W_n|^p$ to zero.