The probability generating functions of a random variable $Y$ is defined by
$$
\phi_Y(t)= \mathbb E[t^Y].
$$
Let $X_n$ and $X$ be non negative random variables such that $X_n$ converges to $X$ in distribution. I know that on $[0,1]$, the probability generating functions also converge point wise
$$
\phi_{X_n}(t) \to \phi_{X}(t), \qquad t \in [0,1].
$$
But I read somewhere that
$$
\max_{x \in [0,1-\epsilon]} |\phi_{X_n}(t) \to \phi_{X}(t)| \to 0.
$$
In other words, the convergence is also uniform on $[0,1-\epsilon]$. Why is this?
Best Answer
Let $p_n$ and $p$ denote the prob. mass functions of $X_n$ and $X$, respectively. We know that $p_n(x)\to p(x)$ and $|p_n(x)-p(x)|\le 1$ for each $x\ge 0$. Then for $0<\delta<1$, \begin{align} \sup_{0\le t\le \delta}|\phi_{X_n}(t)-\phi_{X}(t)|&\le \sum_{0\le x \le M}|p_n(x)-p(x)|+\sup_{0\le t\le \delta}\frac{t^M}{1-t} \\ &=\sum_{0\le x \le M}|p_n(x)-p(x)|+\frac{\delta^M}{1-\delta}. \end{align} Taking $M\ge \ln(\epsilon(1-\delta))/\ln(\delta)$ for some $\epsilon>0$, one gets $$ \limsup_{n\to\infty}\sup_{0\le t\le \delta}|\phi_{X_n}(t)-\phi_{X}(t)|\le\epsilon. $$