Mixed Poisson process

In probability theory, a mixed Poisson process is a special point process that is a generalization of a Poisson process. Mixed Poisson processes are simple example for Cox processes.

Definition

Let

μ

{\displaystyle \mu }

be a locally finite measure on

S

{\displaystyle S}

and let

X

{\displaystyle X}

be a random variable with

X

≥

0

{\displaystyle X\geq 0}

almost surely.

Then a random measure

ξ

{\displaystyle \xi }

on

S

{\displaystyle S}

is called a mixed Poisson process based on

μ

{\displaystyle \mu }

and

X

{\displaystyle X}

iff

ξ

{\displaystyle \xi }

conditionally on

X

=

x

{\displaystyle X=x}

is a Poisson process on

S

{\displaystyle S}

with intensity measure

x

μ

{\displaystyle x\mu }

.

Comment

Mixed Poisson processes are doubly stochastic in the sense that in a first step, the value of the random variable

X

{\displaystyle X}

is determined. This value then determines the "second order stochasticity" by increasing or decreasing the original intensity measure

μ

{\displaystyle \mu }

.

Properties

Conditional on

X

=

x

{\displaystyle X=x}

mixed Poisson processes have the intensity measure

x

μ

{\displaystyle x\mu }

and the Laplace transform

L

(

f

)

=

exp

⁡

(

−

∫

1

−

exp

⁡

(

−

f

(

y

)

)

(

x

μ

)

(

d

y

)

)

{\displaystyle {\mathcal {L}}(f)=\exp \left(-\int 1-\exp(-f(y))\;(x\mu )(\mathrm {d} y)\right)}

.

Sources

Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer.