= Counting process
{title2=$N(t)$}
{wiki}
A <counting process> records accumulated <events>. It is a nonnegative integer-valued <Stochastic process> $N_t$, usually <adapted process>[adapted] and <càdlàg function>[càdlàg], with $N_0=0$ and <monotone function>[nondecreasing] <sample path>[sample paths]. Nonexplosion <means> $N_t<\infty$ at every finite time. In a <simple counting process>, every jump is one; broader counting conventions also allow several <events> at the same instant.
A rate-$\lambda$ <Poisson process> has <independent increments>[independent stationary increments], <mean> $\lambda t$ and <variance> $\lambda t$. A one-event process $N_t=\mathbf1_{\{T\leq t\}}$, with $T>0$, instead records whether its single <event> has occurred. It is a <counting process> without the <Poisson process>'s unrestricted repeated arrivals.
The path $N_t$ is the observed count. A <counting-process intensity> describes its <conditional probability>[conditional] <event> rate, and a <compensator of a counting process>[compensator] accumulates that <predictable process>[predictable] rate or, more generally, a <predictable process>[predictable] <event> measure. These objects are not the same: the count has <integer> jumps, whereas a continuous <compensator of a counting process>[compensator] can increase between observed <events>. In <Survival analysis>, the one-event convention naturally combines a <Hazard function> with an <at-risk process>.
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