Solution
= Solution
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 as
$$
\boxed{U_t\Rightarrow N(0,1)\quad(t\to\infty).}
$$
This is the centered Poisson normal limit with its exact <variance> scaling.