On Minkowski sum of two independent Poisson point processes

Question

Suppose that $\Phi_1$ and $\Phi_2$ represents two independent Poisson point processes respectively with intensity $\lambda_1$ and $\lambda_2$ (therefore). We know very well different operations on Poisson processes preserving the Poisson law like superposition, dilation and thinning.

I would like to know if there is any work on the law of Minkowski sum of two Poisson processes. By Minkowski sum of $\Phi_1=\sum \delta_{x^1_i}$ and $\Phi_2=\sum \delta_{x^2_i}$, I mean a new Point processes like: $\Phi_3=\displaystyle\sum_{i,j} \delta_{x^1_i+x^2_j}$.

I could not find any related literature on this topic, except works on germ-grain model, so any help is appreciated.

Answer 1

Indeed the process is not locally finite if your processes are homogeneous on $R^d$ (that can be deduced from the computations below).

If on the other hand your intensity measures $\lambda_1$ and $\lambda_2$ are finite, then you have a point process that is not Poisson.

Let $K$ be some compact set. Then $$\Phi_3(K)=\sum_{x\in \Phi_1}\Phi_2(K-x).$$ Using Mecke formula yields that $$ E \Phi_3(K)=\int \lambda_1(dx) E(\Phi_2(K-x))=\int \lambda_2(K-x)\lambda_1(dx),$$ and therefore for $K,L$ disjoint $$ E\Phi_3(K)E\Phi_3(L)=\int \int \lambda_2(K-x)\lambda_2(L-y)\lambda_1(dx)\lambda_1(dy)$$ Similar computations yield $$E\Phi_3(K)\Phi_3(L)=\int\int E \Phi_2(K-x)\Phi_2(L-y) \lambda_1(dx)\lambda_1(dy)$$ For any $K,L,x,y,$ writing $$\Phi_2(K-x)=\Phi_2((K-x)\cap(L-y))+\Phi_2((K-x)\setminus(L-y))$$, idem for $\Phi_2(L-y)$, and developing the product yields that in non trivial cases for well chosen $L,K$, $$E\Phi_3(K)\Phi_3(L)>E\Phi_3(K)E\Phi_3(L)$$