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

.

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

Printable View

- August 29th 2009, 06:04 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 ? - August 29th 2009, 06:33 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 ? - August 30th 2009, 03:56 AMfunnyinga
- August 30th 2009, 04:43 AMMoo
Yes, exactly (Nod)

converges weakly towards a uniform distribution. - August 31st 2009, 05:26 AMfunnyinga
Im now only having trouble picturing what happens when . Can anyone please explain?

- September 18th 2009, 07:25 AMMoo
There is no point finding the convergence of . I mean it's not useful for the convergence of random variables here.