Suppose a compound Poisson process is defined as $X\_{t} = \sum\_{n=1}^{N\_t} Y\_n$, where $\{Y\_n\}$ are i.i.d. with some distribution $F\_Y$, and $(N\_t)$ is a Poisson process with parameter $\alpha$ and also independent from $\{Y\_n\}$.

1. Is it true that as 
$t\rightarrow \infty, \, \frac{X\_{t}-E(X\_{t})}{\sigma(X\_t)
 \sqrt(N\_t)} \rightarrow \mathbf{N}(0, 1)$ in distribution, where the limit is a standard Gaussian distribution? I am considering
using Central Limit Theorem to show it, but the
theorem I have learned only applies when $N\_t$ is
fixed and deterministic instead of
being a Poisson process.
2. A side question: is it possible to derive the
distribution of $X\_{t}$, for each $t\geq 0$? Some book that has the derivation?

Thanks!