Suppose X_n (for each positive integer n) is a stochastic variable concentrated on the nonnegative integers. Suppose that the probability generating function P_{X_n}(t)= E(t^(X_n)) of X_n converges pointwise on the closed unit interval to a function g: [0,1] -> R. Under what conditions is g itself a probability generating function? Is it sufficient that g(t) is continuous from the left in t=1?
In other words, I am looking for the pgf-analogue of the continuity theorem of Levy-Cramer which says that if the characteristic function of X_n (not necessarily concentrated on nonnegative integers in this theorem) converges pointwise to a function g (on R) and if g is continuous in 0, then g is itself a characteristic function. Thanks a lot in advance!
$Et^{X_n}$ is same as $Ee^{-sX_n}$ where $s=-\log \, t$. Thus what you are asking for is a continuity theorem for Laplace transforms. This theorem is proved in Volume 2 of Feller's book. The condition is indeed continuity of $g$ from the left at $1$.