Cox process

In

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

${\displaystyle \mu }$.

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

Laplace transform

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 ${\displaystyle f}$.

See also

• 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

References

Notes

Bibliography

• ISBN 0-412-21910-7
• Donald L. Snyder and Michael I. Miller Random Point Processes in Time and Space Springer-Verlag, 1991
  ISBN 3-540-97577-2
  (Berlin)

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