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

, hence if is my random variable

then the standard deviation of that random variable should converge to . I.e.

and, since it is a Poission process

for large N as well.

But then of course

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

?

Is there prove?