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