How to prove a statistic is not complete

Question

Suppose X is a Poisson($\lambda$), where $\lambda\in\{0,1,2,...\}$, how to prove X is not complete. It seems like that we need to find a function $g$ which is not identical $0$, such that the following linear system is always true $$ \sum\limits_{k=0}^{\infty}\frac{\lambda^k}{k!}g(k)=0\quad \forall \lambda\in\{0,1,2,...\} $$ But how to construct such a $g$? Thanks a lot!

Answer 1

The expression $$ f_g(\lambda) = \sum_{k\geq0} \frac{\lambda^k}{k!}g(k) = \sum_{k\geq 0}\lambda^k a_k $$ defines a function $f_g$ of $\lambda$ by a power series around $\lambda=0$. The condition imposed on it is that it vanishes at all nonnegative integers. One way to satisfy this condition is find a function $f(\lambda)$ that vanishes at $\lambda=0$ and is periodic with period $1$, such as $$ f(\lambda) = \sin2\pi\lambda. $$ It's trivial to work out the values $g(k)$ and all the rest.

The condition that $\lambda$ is a nonnegative integer is important here. If $\lambda$ could be anywhere on an interval surrounding the origin, there would be no such function $f$ because any such complex-analytic function would have all its derivatives at the origin equal to zero and would itself be identically zero.