I'm revising for an exam and have no idea how to approach this question:
Let $\{N_t\}_{t\geq 0}$ be a homogeneous Poisson process of parameter $\lambda > 0$. Let $\{X_k\}_{k\geq 0}$ be a sequence of random variables, identically independently distributed, with $E[X_k] = 0$ and $Var[X_k] = \sigma^2 < +\infty$. Is the process defined by $Y_t = X_{N_t}$ strictly stationary?
I'm guessing the fact that the Poisson process has independent increments and we have:
$P(N_{t_2} - N_{t_1} = k) = \dfrac{\lambda (t_2 - t_1))^k}{k!} e^{-\lambda (t_2 - t_1)}$ is important, but not sure how it relates, any hints appreciated!
