Lindeberg's Condition

oercim · science forum beginner Joined: 04 May 2005 Posts: 40

Hi. I am trying to understand Lindeberg-Feller Central Limit
theorem.
This theorem states that: Let X1,X2, . . . , Xn be independent
(but not necessarily indentically distributed) random variables with
E[Xi] = Mi and var(Xi) = si^2<oo;.
Define M =(M1+M2+..........+Mn)/n and
S^2=(s1^2+s2^2+....+s3^2)/n
Then

sqrt(n)*(¯X-M)---d---->N(0,S^2) where d refers converges in
distribution

if Lindenberg Condition is satisfied.

Here is a link that tell Lindeber's Condition:

http://mathworld.wolfram.com/LindebergCondition.html

I can understand the theorem what says except the Lindeberg
Condition.I dont understand what does this condition say? According to
some sources it says the sum of the variances shouldn't be dominated by
one or a few variances of related variables.But how does this condition
says this. I am quite familier with such inequalities but i live
difficulties explaining this condition.
I know this was a long question If u help, I will
appreciative.Thanks.

Özgür Ercim(METU)

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