Definition
Let be a locally finite measure on and let be a random variable with almost surely.
Then a random measure on is called a mixed Poisson process based on and iff conditionally on is a Poisson process on with intensity measure .
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.
Let be a locally finite measure on and let be a random variable with almost surely.
Then a random measure on is called a mixed Poisson process based on and iff conditionally on is a Poisson process on with intensity measure .
Mixed Poisson processes are doubly stochastic in the sense that in a first step, the value of the random variable is determined. This value then determines the "second order stochasticity" by increasing or decreasing the original intensity measure .
Conditional on mixed Poisson processes have the intensity measure and the Laplace transform
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Many stochastic processes can be represented by time series. However, a stochastic process is by nature continuous while a time series is a set of observations indexed by integers. A stochastic process may involve several related random variables.
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Let be some measure space with -finite measure . The Poisson random measure with intensity measure is a family of random variables defined on some probability space such that
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