Cox process

Let ${\displaystyle \xi }$ be a random measure.

A random measure ${\displaystyle \eta }$ is called a Cox process directed by ${\displaystyle \xi }$ , if ${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )}$ is a Poisson process with intensity measure ${\displaystyle \mu }$ .

Here, ${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )}$ is the conditional distribution of ${\displaystyle \eta }$ , given ${\displaystyle \{\xi =\mu \}}$ .

If ${\displaystyle \eta }$ is a Cox process directed by ${\displaystyle \xi }$ , then ${\displaystyle \eta }$ has the Laplace transform

${\displaystyle {\mathcal {L}}_{\eta }(f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\mathrm {d} x)\right)}$

for any positive, measurable function ${\displaystyle f}$ .

• Poisson hidden Markov model
• Doubly stochastic model
• Inhomogeneous Poisson process, where λ(t) is restricted to a deterministic function
• Ross's conjecture
• Gaussian process
• Mixed Poisson process

Notes

Bibliography

• Cox, D. R. and Isham, V. Point Processes, London: Chapman & Hall, 1980 ISBN 0-412-21910-7
• Donald L. Snyder and Michael I. Miller Random Point Processes in Time and Space Springer-Verlag, 1991 ISBN 0-387-97577-2 (New York) ISBN 3-540-97577-2 (Berlin)