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edited Jan 15, 2021 at 9:18

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Question about the intensity of a cox process, Diggle et. alDiggle–Moraga–Rowlingson–Taylor (2013)

On page 2 of "Spatial and Spatio-Temporal Log-Gaussian Cox Processes: Extending the Geostatistical Paradigm""Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle et al.Diggle–Moraga–Rowlingson–Taylor (2013), accessible at https://arxiv.org/pdf/1312.6536.pdf arXiv, they claim the following on the bottom right of the page:

"...in the stationary case the intensity of the Cox process is equal to the expectation of Λ(x)"

My assumption is that by "intensity", they are referring to some extension of $$ \lambda (t)=\lim _{h\downarrow 0}{\frac {1}{h}}\mathbb {E} [N(t+h)-N(t)|{\mathcal {F}}_{t}],$$ as taken from wikipedia at https://en.wikipedia.org/wiki/Intensity_of_counting_processes.

I mentioned "extension" because in Diggle et al'sthe Diggle–Moraga–Rowlingson–Taylor paper, they define the Cox process on $\mathbb {R}^2$, whereas the intensity definition from Wikipedia is to do with processes defined on $\mathbb{R}$.

My issue is that I'm unable to find a proof of their claim. If someone more experienced in this field could explain to me why the authors' claim is true / guide me to a relevant source, I would really appreciate it.

Thank Thank you.

Question about the intensity of a cox process, Diggle et. al (2013)

On page 2 of "Spatial and Spatio-Temporal Log-Gaussian Cox Processes: Extending the Geostatistical Paradigm" by Diggle et al. (2013), accessible at https://arxiv.org/pdf/1312.6536.pdf , they claim the following on the bottom right of the page:

"...in the stationary case the intensity of the Cox process is equal to the expectation of Λ(x)"

I mentioned "extension" because in Diggle et al's paper, they define the Cox process on $\mathbb {R}^2$, whereas the intensity definition from Wikipedia is to do with processes defined on $\mathbb{R}$.

Thank you.

Question about the intensity of a cox process, Diggle–Moraga–Rowlingson–Taylor (2013)

On page 2 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the following on the bottom right of the page:

"...in the stationary case the intensity of the Cox process is equal to the expectation of Λ(x)"

I mentioned "extension" because in the Diggle–Moraga–Rowlingson–Taylor paper, they define the Cox process on $\mathbb {R}^2$, whereas the intensity definition from Wikipedia is to do with processes defined on $\mathbb{R}$.