Weighted moments for the limit of a normalized supercritical Galton- Watson process
Résumé
Let $(Z_n)$ be a supercritical Galton-Watson process, and let $W$ be the limit of the normalized population size $Z_n/m^n$, where $m=\E Z_1>1$ is the mean of the offspring distribution. Let $\ell$ be a positive function slowly varying at $\infty$. Bingham and Bingham and Doney (1974) showed that for $\alpha >1$ not an integer, $\E W^{\alpha}\ell(W) <\infty $ if and only if $\E Z_1^{\alpha}\ell(Z_1) <\infty $; Alsmeyer and Rösler (2004) proved the equivalence for $\alpha>1$ not a dyadic power. Here we prove it for all $\alpha>1$.