Consider the following setup. We are given two $[0,1]$-marked point processes on $\mathbb R$ or $[0,\infty)$, denoted $X,Y$. Denote that the ground processes (i.e. the point processes ignoring marks) by $X_g,Y_g$. Assume that we know that stationary distributions exist for $X$ and for $Y_g$. Assume furthermore that we can construct $Y$ in a way such that $Y=X+Z$ for another point process $Z$, meaning that $X(\omega)\subset Y(\omega)$ for each realisation $\omega$.
Fix the distributions of $X,Y$. Then I was wondering whether we can do one of the following:
- Take $X$ stationary, and select the distribution of $Z$ in such a way that $Y_g$ is stationary;
- Take $Y_g$ stationary, and then select the decomposition in such a way that $X$ is stationary.
I'm aware that this setup may be a bit vague and overly specific, but hopefully something can be said.
Please let me know if more context or assumptions are needed. Any help is much appreciated.