If $\Phi=$ {$x_n$} is a stationary point process and $\Phi_x=${$x_n+x$} is its translation by $x$. why we have $$P(\Phi\in Y||x)=P(\Phi_x \in Y||0)$$ ? $P(\Phi\in Y||x)$ means the point process have property $Y$ and contains a point at $x$. $0$ is the origin.
why is it $0 \in \Phi$? I think it must be $2x \in \Phi$
Reference:
- Sung Nok Chiu, Dietrich Stoyan, Wilfrid S. Kendall, Joseph Mecke, Gilbert. Stochastic Geometry and Its Applications, 3rd Edition. Page 127.
A point process is stationary if the probability of the point distribution having property $Y$ doesn't change when we shift the process by $x$. So $\Phi$ and $\Phi_x$ have the same distribution. The notation probably means that the point $0\in\Phi$ not $0\in\Phi_x$.
$$P(\Phi\in Y || x)=P(\Phi\in Y | x\in\Phi)=P(\Phi_x\in Y | x\in\Phi_x)=P(\Phi_x\in Y | 0\in\Phi)=P(\Phi_x\in Y || 0)$$