central limit theorem, equivalent statements

[melnick]

If it helps, I think the theorem you're referring to is actually Theorem 4
in the following document:

www.math.mcgill.ca/jaksic/notes/clt.ps

So if anyone can furnish a proof of that you'll be set....

"Mamet1212" <mame...@aol.com> wrote in message
news:20040421201127...@mb-m25.aol.com...
>
> >Subject: central limit theorem, equivalent statements
> >From: crazyte...@yahoo.com (J. Woodward)
> >Date: 4/21/2004 3:28 AM Eastern Daylight Time
> >Message-id: <960ee97d.04042...@posting.google.com>
> >
> >My professor wrote the following on the blackboard (we're studying the
> >CLT and related topics):
> >
> >"Let {X_n,i} be a triangular array of random variables (independent
> >within each row). Suppose
> >
> > (*) given epsilon > 0, sum(over i) E( X_n,i I(|X_n,i| <= epsilon) )
> >---> m
> >(**) given epsilon > 0, sum(over i) Var(X_n,i I(|X_n,i| <= epsilon) )
> >---> s^2 finite
> >(***) max(over i) |X_n,i| ---> 0 in probability,
> >
> >where I(A) is the indicator function of the set A.
> >
> >Then sum(over i) X_n,i ---> N(m,s^2) in distribution." I'm going to
> >call this result R1.
> >
> >The problem is that I haven't been able to convince myself that this
> >follows from (or really is an equivalent formulation of) any of the
> >version of the CLT for triangular independent arrays (probably
> >Lindeberg's) that I'm familiar with.
> >
> >I'm guessing there's no loss of generality if we set m=0, s^2=1. Then
> >I'm acquainted with the following:
> >
> >"Let {X_n,i} be a triangular array of random variables (independent
> >within each row). Suppose
> >
> >(1) E(X_n,i) = 0
> >(2) Sum(i=1,...,k_n) E((X_n,i)^2) = 1
> >(3) Given epsilon > 0, Sum (i=1,...,k_n) E( (X_n,i)^2 I(|X_n,i| >
> >epsilon) ) ---> 0.
> >
> >Then Z_n = Sum(i=1,...,k_n) X_n,i ---> N(0,1) in distribution." I'm
> >going to call this result R2.
> >
> >Can someone help show me how R1 is equivalent to one of the more
> >popular formulations of the CLT for triangular arrays (say R2)? I've
> >tried doing various manipulations of the hypotheses of R1 (i.e.
> >flipping around the indicator functions to "I(|X_n,i| > epsilon)" but
> >I still can't put it in just the right form. please help.....
>
>
> This looks pretty close to the version of Lindeberg's theorem stated in
> Billingsley's book, Probability and Measure (3rd ed), theorem 27.4.
>
> Can someone confirm these two statements are the same??