Let be a sequence of absolutely continuous random variables such that has pdf

.

Show that the sequence converges in distribution. What happens when ?

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- Aug 29th 2009, 07:04 AMfunnyingaConvergence of sequence of continuous random variables
Let be a sequence of absolutely continuous random variables such that has pdf

.

Show that the sequence converges in distribution. What happens when ? - Aug 29th 2009, 07:33 AMMoo
Hello,

to check for the convergence of a sequence of rv, you mustn't see the convergence of the pdf, but the convergence of the mgf, cdf, or characteristic function.

so for example, let's find the cdf :

by using the squeeze (or sandwich) theorem, we can show that

so the limiting cdf is

which is the cdf of a uniform distribution over

looks clear to you ? - Aug 30th 2009, 04:56 AMfunnyinga
- Aug 30th 2009, 05:43 AMMoo
Yes, exactly (Nod)

converges weakly towards a uniform distribution. - Aug 31st 2009, 06:26 AMfunnyinga
Im now only having trouble picturing what happens when . Can anyone please explain?

- Sep 18th 2009, 08:25 AMMoo
There is no point finding the convergence of . I mean it's not useful for the convergence of random variables here.