Cox process ($N\mid \Lambda \sim Poisson\; process(\Lambda )$)

A counting process which, conditional on a random nonnegative locally integrable intensity $\Lambda$, is an Inhomogeneous Poisson process. For $L_t={\int }_0^t\Lambda (s)ds$, conditioning gives $\mathbb{E}N(t)=\mathbb{E}{L}_t$ and $\mathrm{Var}N(t)=\mathbb{E}{L}_t+\mathrm{Var}{L}_t$. Thus finite-variance counts are never underdispersed. Randomness of the intensity may induce dependence between increments which are independent conditionally.

When each parent has a Poisson number of independently displaced offspring, the conditional offspring process has the displayed directing measure. For disjoint target sets, multinomial distribution splitting of a parent's Poisson count gives joint probability generating function $\exp (\mu {\sum }_j\kappa (A_j-z)(s_j-1))$. Multiply over independent parents to obtain independent Poisson counts conditional on the parents. Compactly supported displacement ensures local finiteness for a locally finite parent process.

// Destination: discrete-probability-distribution.bigb

A Cox process can be directed by a random locally finite measure, which need not have a density. Conditional counts in disjoint sets are independent Poisson variables with that measure's masses. Uniform circular offspring displacements give a directing measure supported on random circles; its unconditional mean can still be a multiple of planar Lebesgue measure. A random intensity function is the special case in which the directing measure is absolutely continuous.