the marked Poisson process
Sometimes each event is not just a moment but a moment with a tag attached. A car passing a sensor has a speed; an earthquake has a magnitude; a customer has a purchase amount; a falling raindrop has a size. A marked Poisson process records both the arrival time AND the mark, so each event becomes a pair (time, mark) — a point in time decorated with extra information.
Formally, start with a Poisson process of arrival times, and to each arrival independently attach a mark drawn from a fixed mark distribution (the marks can be numbers, categories, or vectors). The defining feature is that, given the times, the marks are independent of each other and of the timing. The remarkable consequence is the marking theorem: the collection of (time, mark) pairs forms a Poisson process on the larger space of times-by-marks. This unifies several earlier ideas as special cases — if the mark is a 'keep/discard' coin you recover thinning; if the mark is a numeric size and you sum the marks you recover the compound Poisson process; if the mark is a position you get a spatial pattern.
Marking is the flexible, general bookkeeping tool for 'random events that each carry data'. It lets you ask questions like 'what is the rate of earthquakes above magnitude 6?' (answer: thin the process by the probability a mark exceeds 6) or 'what is the total energy released?' (answer: a compound sum over the marks). The same independence caveat as everywhere applies: the clean Poisson conclusions need the marks to be independent of the timing and of each other — if big events cluster in time, the simple marked-Poisson model is the wrong tool.
Earthquakes hit a region as a Poisson process at lambda = 20 per year, each with an independent magnitude. If P(magnitude > 6) = 0.05, then by the marking/thinning theorem big quakes (magnitude > 6) arrive as their own Poisson process at rate 0.05*20 = 1 per year — independent of the smaller ones.
Attach an independent tag to each event; (time, mark) pairs are themselves a Poisson process.
Thinning and the compound process are special cases of marking. The clean results need marks independent of timing and of each other — temporal clustering of large marks breaks the model.