Autocorrelator

Let $(E,{\mathcal {A}},\mu )$ be some measure space with $\sigma$-finite measure $\mu$. The Poisson random measure with intensity measure $\mu$ is a family of random variables $\{N_{A}\}_{A\in {\mathcal {A}}}$ defined on some probability space $(\Omega ,{\mathcal {F}},\mathrm {P} )$ such that

i) $\forall A\in {\mathcal {A}},\quad N_{A}$ is a Poisson random variable with rate $\mu (A)$.

ii) If sets $A_{1},A_{2},\ldots ,A_{n}\in {\mathcal {A}}$ don't intersect then the corresponding random variables from i) are mutually independent.

iii) $\forall \omega \in \Omega \;N_{\bullet }(\omega )$ is a measure on $(E,{\mathcal {A}})$

Existence

If $\mu \equiv 0$ then $N\equiv 0$ satisfies the conditions i)–iii). Otherwise, in the case of finite measure $\mu$, given $Z$, a Poisson random variable with rate $\mu (E)$, and $X_{1},X_{2},\ldots$, mutually independent random variables with distribution ${\frac {\mu }{\mu (E)}}$, define $N_{\cdot }(\omega )=\sum \limits _{i=1}^{Z(\omega )}\delta _{X_{i}(\omega )}(\cdot )$ where $\delta _{c}(A)$ is a degenerate measure located in $c$. Then $N$ will be a Poisson random measure. In the case $\mu$ is not finite the measure $N$ can be obtained from the measures constructed above on parts of $E$ where $\mu$ is finite.

Applications

This kind of random measure is often used when describing jumps of stochastic processes, in particular in Lévy–Itō decomposition of the Lévy processes.

References

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