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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...

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You're right, it doesn't follow:

Let $Z_i(t)$, $i=0,1$ be two independent Poisson processes with parameter $\lambda p$.

For $n\le t<n+1$ 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 $$P(Y_t\le Y_s \text{ whenever }t\le s)=0.$$

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