A theorem about the Poisson Point process.

Question

In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson point process.

The theorem states that if $\Pi$ is a poission point process on $S$ with intensity measure $\mu.$ Let $f:S\rightarrow\lbrack0,\infty)$ be a measurable function. And define $$ Z=\int_{S}f\left( x\right) \Pi\left( dx\right) $$

We have the following,

$$ E\left[ Z\right] =\int_{S}f\left( x\right) \mu\left( dx\right) $$

and if $E\left[ Z\right] <\infty,$ then,

$$ Var\left( Z\right) =\int_{S}f\left( x\right) ^{2}\mu\left( dx\right) $$

How should I proof this? I'm thinking about started the proof by assuming $f$ is a simple function, and apply DCT somehow? Are there any proofs for this theorem?

Answer 1

The proof is given in Proposition 19.5 in Lévy Processes and Infinitely Divisible Distributions by Ken Iti Sato. It goes along the lines:

1. Show that $Z$ follows a Compound Poisson Process with characteristic function $$ \varphi_Y(z):={\rm E}[e^{izY}]=\exp\left(\int_S(e^{izf(x)}-1)\,\mu(\mathrm dx)\right),\quad z\in\mathbb{R}. $$

2. Use the derivatives of the characeristic function to obtain expressions of the first and second moment, i.e. $$ i{\rm E}[Y]=\frac{\mathrm d}{\mathrm dz}\varphi_Y(z)\bigg|_{z=0}=i\int_Sf(x)\,\mu(\mathrm dx) $$ and $$ i^2{\rm E}[Y^2]=\frac{\mathrm d^2}{\mathrm dz^2}\varphi_Y(z)\bigg|_{z=0}=i^2\int_Sf(x)^2\mu(\mathrm dx)+\left(i\int_Sf(x)\,\mu(\mathrm dx)\right)^2. $$

3. Use these expressions to find ${\rm E}[Y]$ and ${\rm Var}(Y)$.