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 $(\mathrm{\Omega },\mathcal{ℱ},\mathrm{P})$ such that

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

ii) If sets $A_1,A_2,\dots ,A_n\in \mathcal{𝒜}$ don't intersect then the corresponding random variables from i) are mutually independent.

iii) $\mathrm{\forall }\omega \in \mathrm{\Omega }{N}_{\bullet }(\omega )$ 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 $X_1,X_2,\dots$, mutually independent random variables with distribution $\frac{\mu }{\mu (E)}$, define $N_{\cdot }(\omega )=\sum _{i=1}^{Z(\omega )}{\delta }_{X_i(\omega )}(\cdot )$ 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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