Poission processes and standard deviation

Hi,

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

$\displaystyle \lambda $, hence if $\displaystyle x_i$ is my random variable

then the standard deviation of that random variable should converge to $\displaystyle \lambda$. I.e.

$\displaystyle \lim_{N \rightarrow \infty}\sum_{i=1}^N (x_i - <x>)^{2}=\lambda$

and, since it is a Poission process

$\displaystyle \lambda = <x>$

for large N as well.

But then of course

$\displaystyle \lim_{N \rightarrow \infty}\sum_{i=1}^N \frac{(x_i - \lambda)^{2}}{\lambda} = 1$

Now, assume you have not one Poission process but M processes. Each with it's own $\displaystyle \lambda_j$. And that the random variable $\displaystyle x_j$ 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

$\displaystyle \lim_{M \rightarrow \infty}\sum_{j=1}^M \frac{(x_j - \lambda_j)^{2}}{\lambda_j} = 1$

?

Is there prove?