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 W^{\alpha}\ell(W) <\infty $; Alsmeyer and R\"osler (2004) proved the equivalence for $\alpha>1$ not a dyadic power. Here we prove it for all $\alpha>1$.