Simple counting process ($N_t$)

A simple counting process counts individual events without simultaneous multiplicities. In the usual convention, $N_0=0$, its sample paths are nondecreasing, integer-valued and càdlàg, every jump discontinuity has size one, and $N_t$ is finite for every finite $t$. Equivalently,where the finite arrival times are strictly increasing and have no finite accumulation point.

Simplicity specifies how arrivals are counted; it does not assert independent increments or a Poisson distribution. Deterministic arrivals at times $1,2,3,\dots$ give a simple counting process, as does a renewal process with strictly positive independent and identically distributed interarrival times. A system that records an entire batch as several events at exactly the same time can have a jump discontinuity larger than one and therefore is not simple. The Poisson process adds particular increment laws to these counting-path properties.