Say there is a Poisson arrival process with rate lambda. The realization of that process is a sequence $X$. Denote $x_t$ as the element of $X$ at time $t$.

Suppose I specify the following rule to select a subset $Y$ of that sequence:
From $t=1$ to $t=\infty$, for each $x_t$ I draw a random value from a uniform distribution between 0 and 1. If the value greater than 0.5 then I have $A=A+1$ with initial $A=0$, and otherwise, $A=A-0.5$. I take that $x_t$ and put into $Y$ only when $A$ is an even number.

My question is: is $Y$ also generated by a Poisson process?