Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M4 / e

Since a Poisson count has mean and variance $\lambda t$, substitute the centering and scaling into its moment generating function:This is finite for every real $\theta$ when $t>0$.

For fixed $\theta$, Taylor expansion at zero gives $e^h-1-h=h^2/2+O(h^3)$. With $h=\theta /\sqrt{\lambda t}$, the logarithm of the moment generating function tends to ${\theta }^2/2$. Thus $M_{U_t}(\theta )\to e^{{\theta }^2/2}$ on a neighborhood of zero. The continuity theorem for moment generating functions identifies the limit asThis is the centered Poisson normal limit with its exact variance scaling.