Theoretical Statistics and Mathematics Unit, ISI Delhi

April 4, 2012 (Wednesday) ,
3:30 PM at Webinar

Speaker:
G. Ghurumuruhan,
Indian Statistical Institute, Delhi

Title:
Convergence theorems for stabilizing functionals of Poisson processes

Abstract of Talk

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.