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

Gerold Alsmeyer; Alex Iksanov; Sergej Polotsky; Uwe Roesler

paper

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

arXiv:0903.3935

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 $\sum_{n \geq 0} e^{an} (W - W_{n})$ 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.