Suppose $_1, _2, … , _$ are independent and identically distributed Poisson random variables, each with mean $$. Prove that $ =exp[−(1/)(_1 + _2 + ⋯ + _)]$ converges in probability to $( = 0) = exp(−)$ ?
Note: If a function $()$ is continuous at $$, then for every $ > 0$, there exists a $ > 0$ such that $| − | < $ implies $|() − ()| < $.
I'm thinking of using Weak law of large Number to prove by finding $E(X_n)$ and $Var(X_n)$ then proceed with WLLN, but I'm not sure with it
I don't know if you want to show that $Xn \to e^{-\lambda}$ almost surely. But this is simple. By the strong law of large numbers (SLLN), see f.i. Chow/Teicher (1978), ch. 5.2 Cor. 2 (Kolmogorov), $Zn \to \mathbb{E}Y_1$ almost surely (a.s.), i.e. $\mathbb{P} (\{\omega \in \Omega \colon \lim_{n \to \infty} Z_n(\omega) = \mathbb{E}Y_1\}) = 1$. Here $\mathbb{E}Y_1 = \lambda \in \mathbb{R}$. It follows that for each continuous function $g \colon \mathbb{R}\to \mathbb{R}$ $g(Zn) \to g(\mathbb{E}Y_1)$ a.s. Here $g(x) = e^{-x}$.