Poisson random measure

A Poisson random measure $N$ with intensity measure $\mu$ assigns independent Poisson random variables to disjoint measurable sets, with $N(A)$ having mean $\mu (A)$. Its compensated version is $\tilde{N}=N-\mu$ on sets of finite intensity.

For nonnegative measurable $H$ and a Poisson random measure of sigma-finite intensity $\rho$, removing the selected point from a sum leaves an independent copy of the original process under the integral. For finite intensity, condition on the Poisson count, represent the points as independent samples from normalized $\rho$, and shift the count index using $n/n!=1/(n-1)!$. For sigma-finite intensity, first select points in a finite-intensity set, condition on the independent outside process and then exhaust the space. Repetition gives the corresponding formula for ordered distinct tuples of points.

A sigma-finite measure on a standard Borel space is the intensity measure of a point process with Poisson counts. Partition the space into sets of finite intensity, draw an independent Poisson count on each piece, and conditional on that count draw independent points from its normalized intensity measure. The sum of their Dirac measures is the required Poisson random measure. Non-atomic intensity gives simplicity; local finiteness of the intensity gives local finiteness of the point configuration.

For deterministic square-integrable $f$ and a Poisson random measure of intensity $\nu$, first define the integral for simple functions on disjoint finite-intensity sets. Independence and the equality of Poisson distribution mean and variance prove the displayed isometry. Completion defines the integral for all $f\in L^2(\nu )$, even when $\int |f|d\nu =\infty$. A small-jump compensated martingale is thereby defined through convergence in Lp, not by subtracting two divergent ordinary integrals.

For nonnegative measurable $f$, independent Poisson counts first prove this formula for simple functions on disjoint measurable sets. Monotone approximation gives the general formula, allowing infinite integrals with $e^{-\infty }=0$. Conversely, the formula evaluated at $f={\sum }_j{u}_j{1}_{A_j}$ for disjoint finite-intensity sets gives the joint Laplace transform of independent Poisson counts, so characterizes the law of a Poisson random measure.

Let $M$ be a Poisson random measure of intensity $\mu$, let $\mu (A)<\infty$, and let $f\ge 0$ be measurable. Apply the Laplace functional of a Poisson random measure to $f+t{1}_A$ and differentiate from the right at $t=0$. On the random side, differentiation is dominated by the integrable random variable $M(A)$; on the intensity side, by $1_A$. This proves the displayed identity when $\int (1-e^{-f})d\mu$ is finite. When this integral is infinite, $e^{-M(f)}=0$ almost surely, and the same identity holds with both sides zero.

For a Poisson random measure on ${\mathbb{R}}^d$ with Lebesgue measure intensity and a nonnegative compactly supported continuous function $f$, put $m_r=\mu (B(0,r))$ and $a_r={\int }_{B(0,r)}{e}^{-f}d\mu$. Continuity gives $a_r/m_r\to e^{-f(0)}$. With $L=\mathbb{E}{e}^{-M(f)}$, the count-weighted Laplace functional of a Poisson random measure and independent inside/outside counts give respectivelyBoth converge to the displayed value. The conditioning is meaningful for every $r>0$; it is not conditioning on a positive-probability point at the origin.

A simple locally finite point measure on $[0,\infty )$ with void probability $e^{-\lambda (A)}$ on finite unions of bounded intervals is a Poisson random measure with Lebesgue measure intensity. Void-event intersections factor, hence the occupation indicators on disjoint cells are independent. The dyadic occupancy approximation of a simple point measure is binomial with rare-event probability $1-e^{-2^{-n}}$ and converges to a Poisson count of mean the interval length. Independence survives joint convergence for disjoint intervals. Extension of the mean measure and the Laplace functional of a Poisson random measure to Borel functions identifies all Borel counts, not just intervals.

For unit-rate Poisson process arrivals $Z_i$, $X_p$ is finite almost surely when $p>1$. There are finitely many positive arrivals before time one; the expected contribution afterwards is ${\int }_1^{\infty }{z}^{-p}dz<\infty$, by the Poisson expectation identity proved on simple functions and extended by monotone convergence. Merging independent processes of rates $1$ and $\rho$ produces rate $1+\rho$; rescaling their arrivals by $1+\rho$ restores unit rate. Hence the sum of their inverse-power contributions has law $(1+\rho {)}^p{X}_p$. In particular two independent unit-rate inverse-square sums add to a variable with law $4X_2$.

For a Poisson random measure on $E$ with finite intensity $\nu (E)$, its integral of a finite-valued measurable mark function is the finite sum of the marks at its random atoms. Its characteristic function is $\mathbb{E}\exp (iu\int fdN)=\exp (\int (e^{iuf}-1)d\nu )$. On a time-mark space with intensity $ds\nu (dz)$ and finite mark intensity, restriction to each bounded time interval has finite intensity and defines a Compound Poisson process. Zero-valued marks do not affect the integral and need not appear in its Lévy measure.