Is this random variable almost surely convergence?

Let $\displaystyle X_n $ be a sequence of random variable with probability density of $\displaystyle f_n(x) = \frac {n}{ \pi (1+n^2x^2) } \ \ \ \ \forall n \geq 1 $

Does $\displaystyle X_n $ converges almost surely?

Proof so far:

Claim: $\displaystyle X_n $ converges to 0 almost surely.

Let $\displaystyle \epsilon > 0 $ and fixed $\displaystyle N \in \mathbb {N} $, I want to show that $\displaystyle \lim _{n \rightarrow \infty } P(|X_n|< \epsilon \ \forall n \geq N) = 1 $,

that meansing proving that $\displaystyle P(X_n< \epsilon \ \forall n \geq N) = 1 $ and $\displaystyle P(X_n > - \epsilon \ \forall n \geq N ) = 1 $

First the first one, we have:

$\displaystyle P(X_n < \epsilon \ \forall n \geq N ) = P(X_N \leq \epsilon ) $ since $\displaystyle X_n$ is decreasing monotonically. And that is the CDF of $\displaystyle X_n $, which is:

$\displaystyle \frac {1}{\pi} [ \arctan (N \epsilon ) + \frac {\pi}{2} ] \rightarrow 1 $ as $\displaystyle n \rightarrow \infty $

Similar, the one will work as well. So is this it? Thanks.

Re: Is this random variable almost surely convergence?

Hello,

It makes no sense to say that $\displaystyle X_n$ decreases monotically. There is no relationship between the behaviour of the sequence of pdf's and the behaviour of the sequence of rv's.

But I have to think further for the problem itself :)

Re: Is this random variable almost surely convergence?

Our professor said that you would have assume they are independent and use borel-contelli theorem for.this

Re: Is this random variable almost surely convergence?

Then he gave you all the elements to solve this...

Consider the events $\displaystyle A_n=\{|X_n|>1/n\}$, prove that $\displaystyle \sum_n P(A_n)=\infty$. By Borel-Cantelli's lemma, and because the events are independent, it lets us say that $\displaystyle P(\limsup_n A_n)=0 \Leftrightarrow P(\liminf_n \{|X_n|<1/n\})=1$.

Then write the liminf with quantifiers and it should give you the almost sure convergence :)