# Sufficient Evidence that a Process is a Poisson Process

My question is more general than what the following problem asks, but I figured that having an example to work with would be helpful.

Here's the problem:

Let $$X = \{X(t), t \ge 0\}$$ be a Poisson process with intensity parameter $$\lambda$$. Suppose each arrival is "registered" with probability $$p$$, independent of other arrivals. Let $$Y = \{Y(t), t \ge > 0\}$$ be the process of "registered" arrivals. Prove that $$Y$$ is a Poisson process with parameter $$\lambda p$$.

Now, I can prove that, for each $$t \ge 0$$, $$Y(t)$$ is a Poisson rv with parameter $$\lambda pt$$ (in fact, there is at least one post here on MSE in which this is proven), but doesn't one also need to show that the process $$Y$$ has independent and stationary increments? Or do these properties necessarily follow from this fact? Because, if so, I'm not seeing how...

Let $$Z_i(t)$$, $$i=0,1$$ be two independent Poisson processes with parameter $$\lambda p$$.
For $$n\le t let $$Y(t)=Z_{I_n}(t)$$ where $$I_n$$ are independent Bernoullis($$1/2$$) say.
Then for each $$t$$, $$Y(t)$$ is Poisson($$\lambda pt$$) but $$Y$$ is not a Poisson process, as for instance $$P(Y_t\le Y_s \text{ whenever }t\le s)=0.$$