Consider a random sample of size $n$ from a Poisson distribution with mean $\lambda$. Define the statistic $$Y_n=\exp\left(-\bar{X}_n\right)$$
- Find a constant $c$ (if it exists) such that $Y_n \xrightarrow{P} c$. Does $Y_n \xrightarrow{a.s.} c$?
- Find the asymptotic normal distribution for (suitably scaled and centered) $Y_n$, i.e. find the sequence of constants $a_n,b_n$ such that $$a_n(Y_n-b_n) \xrightarrow{d} N(0,1)$$ Justify all steps.
How would I approach this problem?
For a) The weak law of large numbers applied on the sequence of the i.i.d. random variables $X_i, i=1,2,\ldots, n$ with mean $$μ=Ε[X_i]=λ, \qquad σ^2=Var(X_i)=λ$$ for all $1\le i \le n$ implies that $$\bar{X}_n \overset{P}\to λ$$ and therefore by the Continuous Mapping Theorem (due to continuity of the function $\exp$) also that $$Y_n=\exp(-\bar{X_n})\overset{P}\to \exp(-μ)=\frac{1}{e^{λ}}=:c$$ Now use the strong law of large numbers to establish that this holds also for the stronger notion of almost sure (a.s.) convergence.
For b) Use the Delta method with $g(x)=\exp(-x)$to obtain that $$\sqrt{n}\left[Y_n-c\right]\overset{d}\to N(0, λe^{-2λ})$$ and therefore that $$\sqrt{\frac{n}{λe^{-2λ}}}\cdot\left[Y_n-c\right]\overset{d}\to N(0,1)$$ which gives you the required values $$a_n=\sqrt{\frac{n}{λe^{-2λ}}}, \qquad b_n=c$$ for each $n \in \mathbb N$.