There is Poisson arrival process $R$, and random shift value $\tau$. Distribution of $\tau$ is unknown, but the range of values that $\tau$ takes is between $0$, and $W$ ( $\tau \in [0,W]$).
Is the shifted process $R(\tau)$ is still Poisson?
My approach to the problem is following: Without loss of generality, assume $\tau$ takes discrete and finite number of values, $N$, between $0$ and $W$, where $\delta \tau = \frac{W}{N}$,$\tau_0$ = 0, $\tau_i = i \delta\tau$. Assume also, $ \tau = \tau_{1}$ with probability $p_1$, and $ \tau = \tau_2$ with probability $p_2$ and so on, and $\sum_{i=0}^{N}p_{i}= 1$. In this case, shifted process $R(\tau)$ is combination of N+1 shifted processes. (i.e. $R(\tau) = R(\tau_0) \cup R(\tau_1) ..\cup R(\tau_N)$. Since Poisson process is stationary, each of the shifted processes are Poisson, and overall process is combination of Poisson processes, so $R(\tau)$ is still Poisson.
If this is true, can we have the same approach when $\tau$ has density (instead of mass) function?