Solution
= Solution
{c}
By the addition rule for independent <Poisson random variables>, $X_n$ has the same distribution as a sum of $n$ independent variables $Z_j\sim\operatorname{Poisson}(\lambda)$. Each has <expectation> and <variance> $\lambda$. The <central limit theorem> therefore gives
$$
\boxed{\frac{X_n-n\lambda}{\sqrt n}\ \xrightarrow{d}\ N(0,\lambda).}
$$
For $\lambda=0$, every variable is zero, and the conclusion is the degenerate normal law with zero variance.