On regularly varying moments for discrete laws

For a discrete law F and large n , we investigate the asymptotic relation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$EX_n^\mu \ell (X_n ) \sim C_\mu (EX_n )^\mu \in (a,b),C_\mu > 0(EX_n \to \infty ),$$ \end{document}, where ℓ(·) is an arbitrary slowly varying function. By a result from [6], it follows that regularly varying moments for Power Series Distributions, generated by an entire function of finite order, satisfy the above relation with Cµ = 1 for each µ ∈ (0, ∞).