I have a Poisson Process with rate $\lambda$ and also a filter which is applied on this process. After first event is issued, during time window $T$, all the following events are filtered. After the filter expires, one event could be issued again and then the filter will be activated immediately again. This scenario will repeat forever.
My question is:
Is this "filtered Poisson Process" still a Poisson Process? If so, how to prove it. Does anybody know there is a name for such problem.
Thanks in advance.
As in André Nicolas's comment, your process is certainly not Poisson. In a Poisson process, the holding times between events are iid $\newcommand{\Exp}{\operatorname{Exp}}\Exp(\lambda)$, and in particular take arbitrarily small values with positive probability, but for your process the holding times are bounded below by $T$.
However, your holding times are still iid, so what you have is still a renewal process. And thanks to the memoryless property of the exponential distribution, the holding times are distributed as $T + \Exp(\lambda)$. So that's a pretty complete description of your "filtered" process.