Originally Posted by

**Laurent** When I wrote that, I litteraly meant that $\displaystyle E[N(2)+N(2,3)|N(2),N(1)]=N(2)+E[N(2,3)]$ holds.

The term $\displaystyle N(2)$ should be obvious: it is a function of the variable we are conditioning by, therefore the conditional expectation acts trivially. (Conditioning by $\displaystyle N(1),N(2)$ is like conditioning by the $\displaystyle \mathbb{N}^2$-valued random variable $\displaystyle (N(1),N(2))$, or by the sigma-algebra of events depending on N(1) and N(2))

The second term is because $\displaystyle N(2,3)$ is independent of $\displaystyle (N(1),N(2))$ (it is even independent of $\displaystyle (N(t))_{0\leq t\leq 2}$). There is no mutual independence between $\displaystyle N(2,3),N(1),N(2)$: it would mean that $\displaystyle N(1),N(2)$ are independent...

As you see, these are the same arguments as for 1 random variable.