Let \(X_{1}, \ldots, X_{n}\), be a sequence of i.i.d. random variables that follow the Poisson distribution with mean \(\mu\). Then the sample average \(\bar{X}_{n}=\frac{X_{1}+\cdots+X_{n}}{n}\) is a consistent estimate of \(\mu\) and \(\bar{X}_{n}^{2}\) is a consistent estimate of \(\mu^{2}\).

(a) Determine the asymptotic distribution of \(\bar{X}_{n}\).

(b) Determine the asymptotic distribution of \(\bar{X}_{n}^{2}\).