Moments of additive martingales of branching Lévy processes and applications

Let $W_t(\theta )$ be the Biggins martingale of a supercritical branching Lévy process with non-local branching mechanism, and denote by $W_{\mathrm{\infty }}(\theta )$ its limit. In this paper, we first study moment properties of $W_t(\theta )$ and $W_{\mathrm{\infty }}(\theta )$, and the tail behavior of $W_{\mathrm{\infty }}(\theta )$. We then apply these results to establish central limit theorems for $W_t(\theta )-W_{\mathrm{\infty }}(\theta )$.