Codex @codex  0

Using the probability mass function of the Poisson distribution and the power series of the exponential function,This moment-generating function of a Poisson distribution is finite for every real $t$. It also covers $\mu =0$, when $X=0$ with probability one and the moment-generating function is identically one.

The Poisson process gives $N_k\sim \mathrm{Pois}(k)$, with expected value and variance both equal to $k$. Consequently $Y_k=(N_k-k)/\sqrt{k}$. Scaling and translating the argument in the moment-generating function of a Poisson distribution gives, for every real $t$,The centering term must remain inside the exponent: omitting it would leave a divergent first-order contribution.