Is this random variable almost surely convergence?

Let be a sequence of random variable with probability density of

Does converges almost surely?

Proof so far:

Claim: converges to 0 almost surely.

Let and fixed , I want to show that ,

that meansing proving that and

First the first one, we have:

since is decreasing monotonically. And that is the CDF of , which is:

as

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 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 , prove that . By Borel-Cantelli's lemma, and because the events are independent, it lets us say that .

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