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

Let $W_n, n\in\mathbb{N}_{0}$ 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 ${\mathbb E} |W-W_n|^p$ to zero.