first, this is my first post here. So I apologize in advance for all wrongdoings.

Here is my problem:

I have a random process that follows a Poisson distribution with the parameter
\lambda , hence if x_i is my random variable
then the standard deviation of that random variable should converge to \lambda. I.e.
\lim_{N \rightarrow \infty}\sum_{i=1}^N (x_i - <x>)^{2}=\lambda
and, since it is a Poission process
\lambda = <x>
for large N as well.

But then of course
\lim_{N \rightarrow \infty}\sum_{i=1}^N \frac{(x_i - \lambda)^{2}}{\lambda} = 1

Now, assume you have not one Poission process but M processes. Each with it's own \lambda_j. And that the random variable x_j now is a single realisation of each of those processes.

Is it true (and I have strong reasons to believe it should be, at least under certain circumstances), that
\lim_{M \rightarrow \infty}\sum_{j=1}^M \frac{(x_j - \lambda_j)^{2}}{\lambda_j} = 1

Is there prove?