on the Poisson space

LJK, Grenoble University -- 24 June 2014

From the railway station, take Tram B (direction Gières - Plaine des sports) and get off at the stop Bibliothèques Universitaires. The building you are looking for is the highest one in your back. See here for a map of the campus site.

- 11:30-12:15 Laurent Decreusefond,
Functional Poisson convergence via Stein method.

Consider a Poisson or a binomial process on some state space and a symmetric function f acting on k--tuples of its points. This induces a point process on the target space of f. If k=1, the image is again a Poisson or a binomial process, a property which is not preserved for k ≥ 2. We establish a functional limit theorem, providing an upper bound for an optimal transportation distance between the image process and a suitable Poisson process on the target space for general k. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamic representation for the Poisson process and the Malliavin formalism. The general theory is applied to U--statistics as well as to geometric random graphs, random polytopes and proximity functionals of non--intersecting flat processes. - 12:15-14:00 Lunch
- 14:00-14:45 Dominic Schuhmacher,
Stein's method for Gibbs process approximation in the
total variation metric.

We consider Gibbs point processes on a compact metric space. Our goal is to derive upper bounds for the total variation distance between a general Gibbs process distribution and one that satisfies a certain stability condition. In order to achieve this, we adapt Stein's method to the Gibbs process setting. We re--express the total variation distance in terms of the generator of a spatial birth--death process and reduce the problem to finding an upper bound of the expected coupling time between two spatial birth--death processes with identical transition kernels but started at different configurations. The later problem is solved by constructing an explicit coupling and bounding the time until the two processes meet by the hitting time at zero of a (non--spatial) birth--death process. We obtain upper bounds of the form of an explicit constant times an L_{1}-type distance between the conditional intensities of the point processes. As applications we consider the total variation distance between two pairwise interaction processes and the hard core process approximation of an area--interaction process with small interaction parameter γ. - 14:45-15:30 Anthony Réveillac, Statistical inference for Poisson channels: a Malliavin calculus approach.

The aim of this talk is to present some statistical estimation results in the context of infinite dimensional Poisson channels using the Malliavin calculus. These channels differ from their Gaussian counterpart in several aspects and Malliavin calculus brings, in that setting, new perspectives. For example it allows one to derive a new proof of a remarkable formula in Information Theory, relating the derivatives of the input--output mutual information of a general Poisson channel and the conditional mean estimator of the input, this holding regardless the distribution of the latter. Furthermore, Malliavin calculus offers a universal framework for studying a quite general class of channels. Finally, we will discuss some open questions in that setting. - 15:30-:16:00 Coffee break
- 16:00-16:45 Raphaël Lachièze-Rey, Different scaling regimes for Poisson U--statistics.

Recent results combining Malliavin calculus and Stein's method triggered a series of limit theorems for geometric functionals on the Poisson space. The applications include statistics on the boolean model, intersections of flat processes, estimation of the Sylvester constant, subgraph counting, or the Voronoi approximation of a set, with Gaussian, Poisson, or Gamma limit. We will present some of these results for the functionals that can be put under the form of a Poisson U--statistics, or equivalently for finite sums of multiple integrals with respect to the Poisson measure. It appears that a crucial feature is the area of interaction of the U--statistics, i.e. the average number of points interacting with a typical point. It appears that in many cases the limit law and speed of convergence magnitude depend on this quantity, characterising the scaling regime of the functional. - 16:45-17:30 Jean-François Coeurjolly, Stein's estimation of the intensity of a stationary spatial Poisson point process.

We revisit the problem of estimating the intensity parameter of a homogeneous Poisson point process observed in a bounded window of R^{d}making use of a (now) old idea of James and Stein. For this, we prove an integration by parts formula for functionals defined on the Poisson space. This formula extends the one obtained by Privault and Réveillac (Statistical inference for Stochastic Processes, 2009) in the one-dimensional case. As in Privault and Réveillac, this formula is adapted to a notion of gradient of a Poisson functional satisfying the chain rule, which is the key ingredient to propose new estimators able to outperform the maximum likelihood estimator (MLE) in terms of the mean squared error.

The new estimators can be viewed as biased versions of the MLE but with a well--constructed bias, which reduces the variance. We study a large class of examples and show that with a controlled probability the corresponding estimator outperforms the MLE. We will illustrate in a simulation study that for very reasonable practical cases (like an intensity of 10 or 20 of a Poisson point process observed in the euclidean ball of dimension between 1 and 5) we can obtain a relative (mean squared error) gain of 20% of the Stein estimator with respect to the maximum likelihood.

- Gilles Bonnet (University of Osnabrueck)
- Sébastien Darses (Unisersité Aix-Marseille)
- Laurent Decreusefond (Telecom ParisTech)
- Vincent Douchez (INRIA Grenoble)
- Rémy Drouihlet (Université de Grenoble)
- Pierre Etoré (Université de Grenoble)
- Gérard Grégoire (Université de Grenoble)
- Céline Labart (Université de Savoie)
- Raphaël Lachièze-Rey (Université Paris Descartes)
- Ester Marriuci (Université de Grenoble)
- Nadia Morsli (Université de Grenoble)
- Anthony Réveillac (Université Paris Dauphine)
- Raphaël Rossignol (Université de Grenoble)
- Dominic Schuhmacher (Georg-August-Universität Göttingen)