Understanding defn. of compound poisson processes

Question

I am reading about what seems to be called compound Poisson distributions in Kallenberg's Foundations of Modern Probability, edition 3, in chapter 7. He has the following lemma:

Let $\xi$ be a random vector in $R^d$, fix a bounded measure $\nu \ne 0$ on $R^d - {0}$, and let $c = ||\nu||, \nu' = \nu/c$. Then, the following are equivalent:

(i) $\xi =_d X_{\kappa}$, for random walk $X = (X_n)$ in $R^d$ based on $\nu'$, where $\kappa$ is independent of $X$ and Poisson distributed with mean $c$.

(ii) $\xi =_{d} \int x \eta (dx)$, for Poisson process $\eta$ on $R^d - {0}$ with $E(\eta) = \nu$.

Specifically, I thought that, in the integral, $x \in R^d - {0}$ since $\eta$ is a Poisson process on $R^d - {0}$, but then I don't see how the integral itself makes sense. So:

1. How do we interpret the integral above?
2. Why should this be the same process as described in (i)?
3. Does this correspond to the definition of compound Poisson processes on Wikipedia? I don't think so, because that is explicitly stated as a continuous time process, whereas here we have a fixed random variable.

Answer 1

I will try to answer your questions in order:

1. The integral is easier to interpret than you think. Saying that $\eta$ is a Poisson process on $R^d - 0$ only means that there is no point in 0. But in the same sense, we can interpret it as a Poisson process on $R^d$ (without point in 0)! Also, note that $x$ is 0 in 0 anyway, so that it doesn't even matter whether there is a point in 0 or not.
2. Note that (as long as $\nu$ is bounded), saying that $\eta$ is a Poisson process on $R^d$ with measure $\nu$ is the same as taking a Poisson number with mean $\Vert \nu\Vert$ of points, distributed independently with distribution $\nu'$. If we denote this Poisson number by $\kappa$ and enumerate the points $x_i$ from $1$ to $\kappa$ (in some arbitrary order), we get $$ \int x\;\eta(\mathrm d x) = \sum_{x\in \eta} x = \sum_{i=1}^\kappa x_i $$ which is exactly a RW based on $\nu'$.
3. The compound Poisson process described in the Wikipedia article can be viewed as a special case by viewing time as another space dimension. Indeed, taking the notation from there, let $\nu'$ be the distribution of the independent $D_i$ and consider $\nu(\mathrm d t, \mathrm d x) := 1_{[0,t]}\mathrm d t\otimes \nu'$ on $R^2$; the first dimension being time. Then, $\Vert \nu\Vert = t$ and we obtain the compound Poisson process.