= Simple counting process
{title2=$N_t$}
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,
$$
N_t=\sum_{k\geq1}\mathbf 1_{\{T_k\leq t\}},
$$
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 $1,2,3,\ldots$ 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.
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