🚀 go-pugleaf

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