Intersection limit calculation

Question

If $(N_t)_{t\geq0}$ is a Poisson process of rate $\lambda$, what would $P\left\{\cap \ _{r=0}^mN_r=2r\right\}$ look like? Is it just multiplication of $r = 1$ to $m$ of $(P\left\{N_r=2r\right\})$? I've never seen this notation and I'm a little confused. I would really appreciate it if someone could show me what the final result would look like.

Answer 1

You have right intuition but make a false conclusion... it's right that actually $$P\left(\bigcap_{r=0}^m \{N_r = 2r\}\right)$$ is meant but in general this is not the same like $$\prod_{r=0}^m P\left(\{N_r = 2r\}\right)$$ because the $N_r$'s are not independent.

Consider that $$\begin{align*}P\left(\bigcap_{r=0}^m \{N_r = 2r\}\right) &= P\left(N_0 = 0, N_1 = 2, N_2 = 4, \ldots, N_m = 2r\right) \\ &= P\left(N_1 - N_0 = 2, N_2 - N_1 = 2, \ldots, N_m - N_{m-1} = 2\right)\\ &= \prod_{n=1}^m P\left(N_n - N_{n-1} = 2\right)\end{align*}$$ because increments are independent.

And the last expression you can calculate very easily