Poisson random variables converging to zero

Question

I need help to prove if $X_n$ are Poisson random variables with parameter $1/n$, then $X_n\xrightarrow{\mathbb P} 0$. What is the standard method to solve this kind of question?

Answer 1

Hint

$$\mathbb P\left\{|X_n|>\varepsilon \right\}\leq \frac{1}{\varepsilon^2 }\mathbb E[X_n^2].$$

Answer 2

an alterntative way is to observe that the MGF limit of your sequence is

$$\lim\limits_{n \to \infty}e^{\frac{1}{n}(e^t-1)}=e^0$$

We immediately recognize a MGF of a degenerate distribution in $x_0=0$

As the sequence converge in distribution to a constant, it converges also in probability

Answer 3

Using Markov's inequality and the fact that the support of a Poisson random variable is non-negative, we have:

$$P(|X_n|\ge \epsilon)\le \frac{\mathbb{E}(|X_n|)}{\epsilon}=\frac{\mathbb{E}(X_n)}{\epsilon}=\frac{1}{n\epsilon}\to 0$$

as $n$ goes to infinity.