Let $N_1(t), t \ge 0$ be a Poisson process with rate $\lambda$, and let $N_2(t), t \ge 0$ be a Poisson process with the unit rate 1. Is the composition $N_1(N_2(t))$ a Poisson process as well?
I tried looking this up but I couldn't find a clear answer if Poisson processes are closed under composition, but if they are, it would be interesting to know (and also what the resulting rate would be).