Prove convergence in probability for n * Poisson variable to zero

[LoadedAnvils]

The problem:

Let $\mu_{n} = \frac{1}{n}$ for $n \in \mathbb{N}$. Let $X_{n} \; \mathtt{\sim} \; \textrm{ Poisson}\left( \lambda_{n} \right)$.

Let $Y_{n} = n X_{n}$. Show that $Y_{n} \xrightarrow{P} 0$.

Work I've done:

I've shown that $X_{n} \xrightarrow{P} 0$ by showing that $\mathbb{P} \left( \left| X_{n} \right| > \epsilon \right) \; \to \; 0$ as $n \to \infty$. (By getting $\epsilon$ and $\delta$ so that the limit definition is satisfied.)

I've also tried to show convergence in quadratic mean (which did not converge to 0) and convergence in distribution (which I could not do).

Relevant equations:

$X_{n} \xrightarrow{P} X$ if for every $\epsilon > 0$, $\mathbb{P} \left( \left| X_{n} - X \right| > \epsilon \right) \; \to \; 0$ as $n \to \infty$.

If the distribution converges in quadratic mean, it converges in probability.

If the distribution converges in distribution and it is a point distribution then it converges in probability.

 

[Valkarie]

Solution:Let \epsilon > 0 . From the definition of convergence in probability, we have that \mathbb{P} \left( \left| X_{n} - 0 \right| > \epsilon \right) \; \to \; 0 as n \to \infty . Since Y_{n} = nX_{n}, it follows that \mathbb{P} \left( \left| Y_{n} - 0 \right| > \epsilon \right) = \mathbb{P} \left( \left| nX_{n} \right| > \epsilon \right) \; \to \; 0 as n \to \infty . Therefore, Y_{n} \xrightarrow{P} 0 .