Suppose I take an exponential random variable $T\sim\mathrm{Exp}(\lambda)$ and create a process that "fires" every $T$ seconds. As far as I know, this is a Poisson process.
Since time windows are independent in Poisson processes, I should be able to uniformly remove samples with probability $p$ (i.e., skip "firing" but leave the time interval intact) and still have a Poisson process. On the other hand, the $p$ I removed is also a Poisson process.
This implies that I can more generally label each sample randomly with one of $n$ labels using a categorical variable over $n$ objects, thereby creating $n$ Poisson processes interleaved in time.
Further, if label $l$ has probability $p_l$, then the interval between occurrences might be $T_l\sim\mathrm{Exp}(p_l\lambda)$. (This is a hand-wavy guess based on the fact that the minimum over $n$ exponential variables gives you another exponential variable plus a categorical variable. The fallacy is assuming that all Poisson processes are exponential.)
Does this sound correct? In particular, I'd like to know if the $n$ processes can be considered independent.
Edit: It appears, at least empirically, that the respective time intervals in the $n$ resulting processes have exponential distributions, but I haven't figured out how to check for independence. I am convinced that they're Poisson processes; it's just a matter of determining if that necessarily means that the intervals are exponentially distributed.
Edit 2: For background, this is useful for real-time simulations because it allows a single thread to generate $n$ exponential timers. It could also be useful for increasing the numerical resolution of the tail of the distribution, e.g., with $\mathrm{Exp}(k\lambda)$ and $\mathrm{Bernoulli}(k^{-1})$ to increase the resolution by a factor of $k$.
Correct. You are describing the thinning of a homogeneous Poisson process. The interarrival times for events in the same category are exponential with intensity $p_i \lambda$, that is to say the rate at which they occur is equal to the product of the rate of the main process and the probability of that event being tagged in that respective category.
It is not difficult to prove it mathematically because it is simply a generalization of the Bernoulli case in which the main process is split into two disjoint sub-processes, to the categorical case in which there may be any positive integer number of categories. We may even use an inductive argument for this generalization; e.g., we thin first with probabilities $p_1$ and $1-p_1$; then $p_2/(1-p_1)$ versus $1 - p_2/(1-p_1)$; etc.
The intervals over which successive events of the same type occur will in general overlap with those intervals for other events; e.g., even with only two types of event, you could observe event times $$\{0.82227, 3.34239, 3.49035, 5.59936, 6.47796, 7.60404, 7.98742, 8.43683, 10.7558, 11.0469\}$$ and the corresponding event types might be $$\{A, B, B, A, B, A, B, B, B, A\},$$ in which case the first $A$-to-$A$ interval completely encloses the first $B$-to-$B$ interval.
To answer your question about the independence of the processes, they are independent in the sense that for any fixed interval of observation, the number of events of type $i$ occurring in that interval does not depend on the number of events of any other type occurring in the same interval.