Poisson point process: Difference between revisions

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===The Mecke equation===

The Mecke equation characterizes the Poisson point process. Let <math>\mathbb{N}_\sigma</math> be the space of all <math>\sigma</math>-finite measures on some general space <math>\mathcal{Q}</math>. A point process <math>\eta</math> with intensity <math>\lambda</math> on <math>\mathcal{Q}</math> is a Poisson point process if and only if for all measurable functions <math>f:\mathcal{Q}\times\mathbb{N}_\sigma\to \mathbb{R}_+</math> the following holds

:<math>\PrE \left[\int f(x,\eta)\eta(\mathrm{d}x)\right]=\int \PrE \left[ f(x,\eta+\delta_x) \right] \lambda(\mathrm{d}x)</math>

For further details see.<ref name="Proper Point Process">{{cite book|author1=Günter Last|author2=Mathew Penrose|title=Lectures on the Poisson Process|url=http://www.math.kit.edu/stoch/~last/seite/lectures_on_the_poisson_process/media/lastpenrose2017.pdf|date=8 August 2017}}</ref>