Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 31 / 1

Write $z=(x,y)$ and $\mu (dz)=\lambda (z)dz$. Treat the nonnegative plant weights as independent marks conditional on the locations, with a measurable mark law $K_z(dw)$ depending only on $z$. Its first and second moments areThis is the independent-marking interpretation appropriate for location-dependent weights.

The general results needed are the following. The Independent marking theorem for Poisson point processes says that such marks turn a Poisson point process of intensity $\mu$ into a Poisson point process on location-mark space with intensity $\nu (dz,dw)=\mu (dz)K_z(dw)$. The Campbell first-moment formula says that for any nonnegative measurable $g$, $\mathbb{E}{\sum }_{u\in \Pi }g(u)=\int gd\nu$, allowing infinite values. The second Poisson factorial moment measure identity says that for nonnegative measurable $h$,These statements apply to sigma-finite intensities and therefore do not require finitely many plants in the entire field. We use these general results without proving them.

Define the total as the nonnegative sum $W={\sum }_{(z,w)\in \Pi }w$, or equivalently as the increasing limit of its finite-intensity truncations. the Campbell theorem givesSince $M<\infty$, the nonnegative random variable $W$ cannot be infinite on an event of positive probability. Thus $W$ is finite almost surely, even if the number of plants is infinite.

Separate the square into its diagonal and ordered distinct-pair terms. Nonnegativity allows the use of Tonelli's theorem and increasing limits, givingThe assumed second integral is finite, so $W$ is square-integrable. Subtracting $(\mathbb{E}W{)}^2$ yieldsThe term $v$ accounts for random weights at fixed locations; the term $m^2$ accounts for the Poisson fluctuations in the locations and count. Nonnegativity is natural for weights. For signed marks, absolute first-moment integrability, rather than just a conditionally convergent integral of the signed mean, would be needed for the ordinary sum.