Suppose $X_1,X_2,…,X_n$ are iid Poisson random variables, each with mean $\theta$. How to prove that $Y_n=\exp[−\frac{1}{n}(X_1+X_2+⋯+X_n)]$ converges in probability to $P(X=0)=\exp(−\theta)$ ?
Hint: If a function $f(z)$ is continuous at $k$, then for every $\epsilon>0$, there exists a $δ>0$ such that $|z−k|<δ$ implies $|f(z)−f(k)|<\epsilon$
I assume $Z_n=\frac{1}{n}(X_1,X_2,…,X_n)$ and used WLLN to prove $Z_n$ converges in probability to $\theta$ for every $\ \epsilon>0$ and I have $X_n=\exp (-Z_n)$ now, but how to use the hint to continue the prove? Do I need to show that $X_n$ is continuous first?
Let $Z_n:=n^{-1}\sum_{i=1}^n X_i$. By the weak law of large numbers, $Z_n\to\mathbb E[X_1]=\theta$ in probability.
The hint suggests to show that if a sequence $(Z_n)$ converges in probability to some $c$ and if $f$ is continuous at $c$, then $\left(f(Z_n)\right)$ converges in probability to $f(c)$. See here for a proof.