SLLN and CLT for dble. index RV ?

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3 total messages Started by Brett Ninness Tue, 14 Apr 1992 22:14

#204

Author: Brett Ninness
Date: Tue, 14 Apr 1992 22:14

42 lines
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I've proved strong convergence of an estimator arising in engineering
problems using the following SLLN:

{X_k} independent, E[X_k] = 0, 

lim_{N\rightarrow\infty}\sum_{k=1}^N\frac{E[X_k^2]}{k^2} < \infty

gives

\frac{1}{N}\sum_{k=1}^M X_k \rightarrow 0 with probability one.

The only twist is that my sequence {X_k} is double indexed: {X_{k,N}}.
That is, the variances of the X_j in

X_1,X_2, ....... X_k

are different to the variances of the X_j in 

X_1,X_2,.........X_k,X_{k+1}.

To put it another way, the variance of a particular X_k depends on how
may terms I have in the sequence, hence the double indexing.  All the terms
are zero mean and independent no matter how long the sequence.

The question is, does this affect my use of the aforementioned SLLN ?.

I also have the same problem with my proof of asymptotic Normality using
Lindberg's Theorem with Lyapunov's condition.

If someone could point me towards some references on SLLN and CLT for 
doubly indexed random variables I would be very grateful.

Brett Ninness
Dept. Elec. Eng.
University of Newcastle.
AUSTRALIA

eemn@cc.newcastle.edu.au

Re: SLLN and CLT for dble. index RV ?

#206

Author: Andrew.Robinson
Date: Wed, 15 Apr 1992 05:34

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In article <1992Apr15.081401.1@cc.newcastle.edu.au> eemn@cc.newcastle.edu.au
(Brett Ninness) writes:
[...]
>The only twist is that my sequence {X_k} is double indexed: {X_{k,N}}.
>That is, the variances of the X_j in
>
>X_1,X_2, ....... X_k
>
>are different to the variances of the X_j in 
>
>X_1,X_2,.........X_k,X_{k+1}.
>
>To put it another way, the variance of a particular X_k depends on how
>may terms I have in the sequence, hence the double indexing.  All the terms
>are zero mean and independent no matter how long the sequence.
>
I'm intrigued.  How can you get a sequence of random variables whose values are
independent, but whose variances depend on the number of values in the
sequence?

regardless,
&rue

Andrew.Robinson@anu.edu.au   <= forestry statistician 
(if it moves, count it; if it doesn't move then CHOP IT DOWN and count it.)
My employers don't care what I say, in fact I don't even think they know I
exist.

Re: SLLN and CLT for dble. index RV ?

#211

Author: USENET News System
Date: Wed, 15 Apr 1992 22:04

28 lines
1435 bytes

In article <1992Apr15.053451.25886@newshost.anu.edu.au>, Andrew.Robinson@anu.edu.au writes:
> 
> In article <1992Apr15.081401.1@cc.newcastle.edu.au> eemn@cc.newcastle.edu.au
> (Brett Ninness) writes:
> [...]
> >X_1,X_2,.........X_k
> >
> >are different to the variances of the X_j in 
> >
> >X_1,X_2,.........X_k,X_{k+1}.
> >
> >To put it another way, the variance of a particular X_k depends on how
> >may terms I have in the sequence, hence the double indexing.  All the terms
> >are zero mean and independent no matter how long the sequence.
> >
> I'm intrigued.  How can you get a sequence of random variables whose values are
> independent, but whose variances depend on the number of values in the
> sequence?
> 
Easily, if k is chosen first, and then the sequence is observed.

However, if you normalise the sequence, and let the normalised sequence be

     Y_1, Y_2, ..., Y_k

can't you apply the CLT to the Y's?

Terry Moore,  Massey University

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