Re: RV and CLT
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 20 Nov 2005 20:58:31 -0500
In article <1132528236.870862.295400@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Wing <nilland@xxxxxxxxxxxxx> wrote:
>Hi,
>I am considering a problem about central limit theorm.
>Given a sequence of random variables {X_n} with E(X_n)=1 and Var(X_n)
>tends to infinity as n grows. Is it possible to find some constants a_n
>and b_n such that (X_n - a_b)/b_n converges to a standard normal
>distribution?
>I think not.
>Could someone have a suggestion? I have no idea to prove or disprove
>it. Thank you.
If b_n goes to infinity so that b_(n+1)/b_n -> 1, a necessary
and sufficient condition is the Lindeberg-Feller condition that
for every e > 0, there exist c_n such that
\sum^n\int_(|x -c_k| <= eb_n) (x -c_k)^2 dF_k(x) /b_n -> 1.
This is needed no matter what, unless some of the X's are close
to normal. Variances going to 0 are as bad a problem.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
- References:
- RV and CLT
- From: Wing
- RV and CLT
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