Functionals of spatial point process often satisfy a weak spatial dependencecondition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which isa level-3 result for empirical point fields as well as a level-2 resultfor empirical point measures. The level-3 rate function coincides withthe so-called specific information. We show that the general resultcan be applied to prove MDPs for various particular functionals,including random sequential packing, birth-growth models, germ-grainmodels and nearest neighbor graphs. 


We provide an alternative algebraic and geometric approach to the results of [I. Bailleul, Probab. Theory Related Fields141 (2008) 283–329] describing the asymptotic behaviour of the relativistic diffusion.

Branching process approximation to the initial stages of an epidemicprocess has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems.One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, insuch a way that their paths coincide for as long as possible. Inthis paper, it is shown, in the context of a Markovian model of parasiticinfection, that coincidence can be achieved with asymptotically high probability until MN infections have occurred, as long asMN = o(N 2/3), where N denotes the total number of hosts.

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: $(A_t^{-}:= \int_0^t 1_{X_s < 0}{\rm d}s, t\geq 0)$ . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung.43 (2006) 171–246]).

We describe quantization designs which lead to asymptotically and order optimal functional quantizers for Gaussian processes in a Hilbert space setting. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions.

This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chainwhich is irreducible and reversible outside the absorbing point.This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues.In the discrete time setting, if the underlying Dirichlet eigenvalues(namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point)are nonnegative, we show that T is distributed as a mixture ofsums of independent geometric laws whose parameters are successiveDirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law.This result leads to a probabilistic interpretation of the spectrum,in terms of strong random times and local equilibria through a simple intertwining relation.Next this study is extended to the continuous time framework,wheregeometric laws have to be replaced by exponential distributions having the (opposite)Dirichlet eigenvalues of the generator as parameters.Returning to the discrete time setting we consider the influence ofnegative eigenvalues which are given another probabilistic meaning.These results generalize results of Karlin and McGregor and Keilson for birth and death chains.

Assume that (Xt )t∈Z is a real valued time seriesadmitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat.24 (1996) 508–539] propose near-minimax estimators $\widehat f_n$ based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators $\widehat f_n$ depend on weak dependence properties of the sequence (Xt )t∈Z through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.

In this article we tackle the problem of inverse non linear ill-posedproblems from a statistical point of view. We discuss the problemof estimating an indirectly observed function, without priorknowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show that their rate of convergence is optimal for the given estimation procedure.

We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

Many statistical applications require establishingcentral limit theorems for sums/integrals $S_T(h)=\int_{t \in I_T} h (X_t) {\rm d}t$ or for quadratic forms $Q_T(h)=\int_{t,s \in I_T} \hat{b}(t-s) h (X_t, X_s) {\rm d}s {\rm d}t$ , where X t is a stationaryprocess. A particularly important case is that of Appellpolynomials h(X t ) = P m (X t ), h(X t ,X s ) = P m , n (X t ,X s ), since the “Appell expansion rank" determines typically thetype of central limit theorem satisfied by the functionalsS T (h), Q T (h). We review and extend here to multidimensional indices, along lines conjectured in [F. Avram and M.S. Taqqu, Lect. Notes Statist.187 (2006) 259–286], a functionalanalysis approach to this problem proposed by [Avram and Brown, Proc. Amer.Math. Soc.107 (1989) 687–695] based on the method of cumulants and on integrabilityassumptions in the spectral domain; several applications arepresented as well.

In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increasedfrequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for “rare events” defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation.

The sequence of random probability measures ν n that gives a path of length n, $\frac{1}{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment.Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.

In a general model (AIMD) of transmission control protocol (TCP)used in internet traffic congestion management, the time dependentdata flow vector x(t) > 0 undergoes a biased random walk ontwo distinct scales. The amount of data of each component x i (t)goes up to x i (t)+a with probability 1-ζ i (x) ona unit scale or down to γ x i (t), 0 < γ < 1 withprobability ζ i (x) on a logarithmic scale, where ζ i depends on the joint state of the system x. Weinvestigate the long time behavior, mean field limit, and the oneparticle case. According to c = lim inf|x|→∞ |x|ζ i (x), the process drifts to ∞ in thesubcritical c < c + (n, γ) case and has an invariantprobability measure in the supercritical case c > c + (n, γ).Additionally, a scaling limit is proved when ζ i (x)and a are of order N –1 and t → N t, in the form of acontinuum model with jump rate α(x).

We consider branching random walks with binary search trees as underlying trees. We show that the occupation measure of the branching random walk, up to some scaling factors, converges weakly to a deterministic measure. The limit depends on the stable law whose domain of attraction contains the law of the increments. The existence of such stable law is our fundamental hypothesis. As a consequence, using a one-to-one correspondence between binary trees and plane trees, we give a description of the asymptotics of the profile of recursive trees. The main result is also applied to the study of the size of the fragments of some homogeneous fragmentations.

We prove a central limit theorem for linear triangulararrays under weak dependence conditions. Our result is then appliedto dependent random variables sampled by a ${\mathbb Z}$ -valued transient random walk. This extends the resultsobtained by [N. Guillotin-Plantard and D. Schneider, Stoch. Dynamics3 (2003) 477–497]. An applicationto parametric estimation by random sampling is also provided.

We study sample-based estimates of the expectation of the functionproduced by the empirical minimization algorithm. We investigate theextent to which one can estimate the rate of convergence of theempirical minimizer in a data dependent manner. We establish threemain results. First, we provide an algorithm that upper bounds theexpectation of the empirical minimizer in a completelydata-dependent manner. This bound is based on a structural resultdue to Bartlett and Mendelson, which relates expectations to sampleaverages. Second, we show that these structural upper bounds can beloose, compared to previous bounds. In particular, we demonstrate aclass for which the expectation of the empirical minimizer decreasesas O(1/n) for sample size n, although the upper bound based onstructural properties is Ω(1). Third, we show that thislooseness of the bound is inevitable: we present an example thatshows that a sharp bound cannot be universally recovered fromempirical data.

We prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables with finite second moment.

Entropic projections and dominating points are solutions to convexminimization problems related to conditional laws of largenumbers. They appear in many areas of applied mathematics such asstatistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugateduality and functional analysis, criteria are derived for theexistence of entropic projections, generalized entropicprojections and dominating points. Representations of thegeneralized entropic projections are obtained. It is shown thatthey are the “measure component" of the solutions to someextended entropy minimization problem. This approach leads to newresults and offers a unifying point of view. It also permits toextend previous results on the subject by removing unnecessarytopological restrictions. As a by-product, new proofs of alreadyknown results are provided.

The estimation of probabilistic deformable template models incomputer vision or of probabilistic atlases in Computational Anatomyare core issues in both fields.A first coherent statistical framework where the geometrical variability ismodelled as a hiddenrandom variable has been given by [S. Allassonnière et al., J. Roy. Stat. Soc.69 (2007) 3–29]. They introduce a Bayesian approach and mixture of them to estimate deformable template models.A consistent stochastic algorithm has been introduced in [S. Allassonnière et al. (in revision)] to face the problem encountered in [S. Allassonnière et al., J. Roy. Stat. Soc.69 (2007) 3–29] for theconvergence of the estimation algorithm for the one component model inthe presence of noise.We propose here to go on in this direction of using some “SAEM-like”algorithm to approximate the MAP estimator in the general Bayesian setting ofmixture of deformable template models.We also prove the convergence of our algorithm toward a criticalpoint of the penalised likelihood of the observations andillustrate this with handwritten digit images and medical images.

We consider in this paper the statistical linear inverse problem Y = Af + ϵ ξ where A denotes a compact operator, ϵ a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y. We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators.