Question about convergence in distribution to an a.s null random variable.

Question

Let $X_{n}\sim \operatorname{Bin}(n,p_{n})$. Suppose $np_{n}\underset{n\to \infty}{\longrightarrow}\lambda, \lambda\in \mathbb{R}$. Given that $\mathbb{P}(X_{n}=k)\underset{n\to \infty}{\longrightarrow}e^{-\lambda}\frac{\lambda^{k}}{k!}$, for $k\in \{0,\ldots,n\}$, show that if $\lambda=0$, then $X_{n}\stackrel{d}{\longrightarrow}N$, where $N$ is an a.s null random variable. My question is: Is it enough to have $\mathbb{P}(X_{n}=k)\underset{n\to \infty}{\longrightarrow}0$?

Answer 1

Let $\varepsilon>0$. If $\lambda=0$, note that $\mathbb{P}(X_{n}=0)=(1-p_{n})^{n}\underset{n\to \infty}{\longrightarrow }e^{-\lambda}=1$ and $\mathbb{P}(X_{n}=k)\underset{n\to \infty}{\longrightarrow}e^{-\lambda}\cdot\frac{\lambda^k}{k!}=0$, for all $k\in \{1,\ldots,n\}$. In this way, we have to: $$\mathbb{P}(|X_{n}|>\varepsilon)=\mathbb{P}(X_{n}>\varepsilon)=\begin{cases}1-\mathbb{P}(X_{n }=0) & \varepsilon\in (0,1) \\ \displaystyle\sum_{j=i+1}^{n}\mathbb{P}(X_{n}=j) & \varepsilon \in[ i,i+1) \text{, }\forall i=1,\ldots,n-1 \\ 0 & \varepsilon\geq n\end{cases}$$ If we take $n\longrightarrow \infty$, we get $\mathbb{P}(|X_{n}|>\varepsilon)\longrightarrow 0$. Which proves that $X_{n}\stackrel{\mathbb{P}}{\longrightarrow}N$, that is, there is convergence in probability. This then implies that $X_{n} \stackrel{d}{\longrightarrow} N$.