DESIGN-ADAPTIVE POINTWISE NONPARAMETRIC REGRESSION ESTIMATION FOR RECURRENT MARKOV TIME SERIES

A general framework is proposed for (auto)regression nonparametric estimation of recurrent time series in a class of Hilbert Markov processes with a Lipschitz conditional mean. This includes various nonstationarities by relaxing usual dependence assumptions as mixing or ergodicity, which are replaced with recurrence. The cornerstone of design-adaptation is a data-driven bandwidth choice based on an empirical bias variance tradeoff, giving rise to a random consistency rate for a uniform kernel estimator. The estimator converges with this random rate, which is the optimal minimax random rate over the considered class of recurrent time series. Extensions to general kernel estimators are investigated. For weak dependent time-series, the order of the random rate coincides with the deterministic minimax rate previously derived. New deterministic estimation rates are obtained for modified Box-Cox transformations of Random Walks.Nonparametric regression estimation, Recurrent time series, Design-adaptation, Optimalrandom estimation rate.

On Spatial Point Processes with Uniform Births and Deaths by Random Connection

This paper is focused on a class of spatial birth and death process of the Euclidean space where the birth rate is constant and the death rate of a given point is the shot noise created at its location by the other points of the current configuration for some response function $f$. An equivalent view point is that each pair of points of the configuration establishes a random connection at an exponential time determined by $f$, which results in the death of one of the two points. We concentrate on space-motion invariant processes of this type. Under some natural conditions on $f$, we construct the unique time-stationary regime of this class of point processes by a coupling argument. We then use the birth and death structure to establish a hierarchy of balance integral relations between the factorial moment measures. Finally, we show that the time-stationary point process exhibits a certain kind of repulsion between its points that we call $f$-repulsion

Optimal estimation for discrete time jump processes

Optimum estimates of nonobservable random variables or random processes which influence the rate functions of a discrete time jump process (DTJP) are obtained. The approach is based on the a posteriori probability of a nonobservable event expressed in terms of the a priori probability of that event and of the sample function probability of the DTJP. A general representation for optimum estimates and recursive equations for minimum mean squared error (MMSE) estimates are obtained. MMSE estimates are nonlinear functions of the observations. The problem of estimating the rate of a DTJP when the rate is a random variable with a probability density function of the form cx super K (l-x) super m and show that the MMSE estimates are linear in this case. This class of density functions explains why there are insignificant differences between optimum unconstrained and linear MMSE estimates in a variety of problems

Quantitative speeds of convergence for exposure to food contaminants

In this paper we consider a class of piecewise-deterministic Markov processes (PDMPs) modeling the quantity of a given food contaminant in the body. On the one hand, the amount of contaminant increases with random food intakes and, on the other hand, decreases thanks to the release rate of the body. Our aim is to provide quantitative speeds of convergence to equilibrium for the total variation and Wasserstein distances via coupling methods.Comment: 20 page

Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks

We study large deviations principles for $N$ random processes on the lattice $\Z^d$ with finite time horizon $[0,\beta]$ under a symmetrised measure where all initial and terminal points are uniformly given by a random permutation. That is, given a permutation $\sigma$ of $N$ elements and a vector $(x_1,...,x_N)$ of $N$ initial points we let the random processes terminate in the points $(x_{\sigma(1)},...,x_{\sigma(N)})$ and then sum over all possible permutations and initial points, weighted with an initial distribution. There is a two-level random mechanism and we prove two-level large deviations principles for the mean of empirical path measures, for the mean of paths and for the mean of occupation local times under this symmetrised measure. The symmetrised measure cannot be written as any product of single random process distributions. We show a couple of important applications of these results in quantum statistical mechanics using the Feynman-Kac formulae representing traces of certain trace class operators. In particular we prove a non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein statistics and mean field interactions. A special case of our large deviations principle for the mean of occupation local times of $N$ simple random walks has the Donsker-Varadhan rate function as the rate function for the limit $N\to\infty$ but for finite time $\beta$. We give an interpretation in quantum statistical mechanics for this surprising result

Universality in a class of fragmentation-coalescence processes

We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non- random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit self-organised criticality in the cluster size distribution, with universal exponent 3/2.Comment: 17 pages, 1 figur