Say there is a Poisson arrival process with rate lambda. The realization of that process is a sequence X$X$. Denote x_t$x_t$ as the element of X$X$ at time t$t$.
Suppose I specify the following rule to select a subset Y$Y$ of that sequence: From t=1$t=1$ to t=infinity$t=\infty$, for each x_t$x_t$ I draw a random value from a uniform distribution between 0 and 1. If the value >0greater than 0.5 then I have A=A+1$A=A+1$ with initial A=0$A=0$, and otherwise, A=A-0.5$A=A-0.5$. I take that x_t$x_t$ and put into Y$Y$ only when A$A$ is an even number.
My question is: is Y$Y$ also generated by a Poisson process?
