Seminar at SMU Delhi

Let $\mathcal{N}$ denote a realization of a Poisson point process in $\mathbb{R}^d$ with intensity measure $\Lambda(.)$ that is comparable to the Lebesgue measure. For a real valued function $f \in \mathcal{L}^p$ that stabilizes at a sufficiently large rate and for a convex compact set $W$, we obtain sharp estimates for the concentration of the sequence $$Y_n = \frac{1}{\Lambda(nW)} \sum_{v \in nW \cap \mathcal{N}} f(v, \mathcal{N})$$ around its mean as $n \to \infty$. As a consequence, we prove almost sure convergence for a large class of functionals of Binomial processes with weakened assumptions. Finally, as an illustration, we show that parameters in Germ-grain models like Voronoi Tessellation and Radial Spanning Tree stabilize at rate $\alpha$ for every $\alpha > 0$ and apply our results to obtain the rate of convergence for the corresponding estimators.