It is proved that each Hoeffding space associated with a random permutation(or, equivalently, with extractions without replacement from a finitepopulation) carries an irreducible representation of the symmetric group,equivalent to a two-block Specht module.

This paper deals with variable selection in regression and binary classificationframeworks. It proposes an automatic and exhaustive procedure which relies on the use ofthe CART algorithm and on model selection via penalization. This work, of theoreticalnature, aims at determining adequate penalties, i.e. penalties whichallow achievement of oracle type inequalities justifying the performance of the proposedprocedure. Since the exhaustive procedure cannot be realized when the number of variablesis too large, a more practical procedure is also proposed and still theoreticallyvalidated. A simulation study completes the theoretical results.

We build confidence balls for the common density s of a real valued sample X1,...,Xn. We use resampling methods to estimate the projection of s onto finite dimensional linear spaces and a model selection procedure to choose an optimal approximation space. The covering property is ensured for all n ≥ 2 and the balls are adaptive over a collection of linear spaces.

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.

In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which one can visualize repetitions of subwords(seen as suffixes of subsequences). The CGR-tree turns a sequence of letters into a Digital Search Tree (DST), obtained from the suffixes of the reversed sequence. Several results are known concerning the height, the insertion depth for DST built from independent successive random sequences having the same distribution. Here the successive inserted words are strongly dependent. We give the asymptotic behaviour of the insertion depth and the length of branches for the CGR-tree obtained from the suffixes of a reversed i.i.d. or Markovian sequence.This behaviour turns out to be at first order the same one as in the case of independent words. As a by-product, asymptotic results on the length of longest runs in a Markovian sequence are obtained.

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.

Let (Sk)k≥1 be the classical Bernoulli random walk on the integer line with jump parametersp ∈ (0,1) andq = 1 − p. The probability distribution of the sojourntime of the walk in the set of non-negative integers up to a fixed time is well-known, butits expression is not simple. By modifying slightly this sojourn time through a particularcounting process of the zeros of the walk as done by Chung & Feller [Proc.Nat. Acad. Sci. USA 35 (1949) 605–608], simpler representationsmay be obtained for its probability distribution. In the aforementioned article, only thesymmetric case (p = q = 1/2) isconsidered. This is the discrete counterpart to the famous Paul Lévy’s arcsine law forBrownian motion.

In the present paper, we write out a representation for this probabilitydistribution in the general case together with others related to the random walk subjectto a possible conditioning. The main tool is the use of generating functions.

We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr -optimal quantizers of an $\mathbb{R}^d$ -valued random vector $X \in L^r(\mathbb{P})$ defined in the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with distribution $\mathbb{P}_{X} = P$ . To be precise, we investigate the Ls -quantization rate of sequences $\alpha_n^{\theta,\mu} = \mu + \theta(\alpha_n-\mu)=\{\mu + \theta(a-\mu), \ a \in \alpha_n \}$ when $\theta \in \mathbb{R}_{+}^{\star}, \mu \in \mathbb{R}, s \in (0,r)$ or s ∈ (r, +∞) and $X \in L^s(\mathbb{P})$ . We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls -rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple (θ*,µ*) such that (αθ*,µ*)n≥1 also satisfies the so-called Ls -empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically Ls -optimal. In both cases the sequence (αθ*,µ*)n≥1 is incredibly close to Ls -optimality. However we show (see Rem. 5.4) that this last sequence is not Ls -optimal (e.g. when s = 2, r = 1) for the exponential distribution.

In this paper, we extend the results of Orey and Taylor [S. Orey and S.J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. 28 (1974) 174–192]relative to random fractals generated by oscillations of Wiener processes to a multivariate framework. We consider a setup where Gaussian processes are indexed by classes of functions.

If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(x)dx < ∞ for some linear functional u and all t ∈ (0,1), then, for each t ∈ (0,1) and all large enough n, the n-fold convolution of the t-tilted density \hbox{$\tilde p_t$}$\begin{array}{}{}_{\mathrm{˜}}\\ {\mathit{p}}_{\mathit{t}}\end{array}$ := e〈x, tu〉p(x)/R(t) is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic functions are useful for saddle-point approximations.

Multitype branching processes and Feller diffusion processes are conditioned on very late extinction. The conditioned laws are expressed as Doob h-transforms of the unconditioned laws, and an interpretation of the conditioned paths for the branching process is given, via the immortal particle. We study different limits for the conditioned process (increasing delay of extinction, long-time behavior, scaling limit) and provide an exhaustive list of exchangeability results.

Using integration by parts on Gaussian spacewe construct a Stein Unbiased Risk Estimator (SURE)for the drift of Gaussian processes, based on theirlocal and occupation times.By almost-sure minimization of the SURE risk ofshrinkage estimators we derive an estimation and de-noisingprocedure for an input signal perturbed by acontinuous-time Gaussian noise.

We study two systems that are based on sums of weakly dependentBernoulli random variables that take values ± 1 with equalprobabilities. We show that already one step of the so-calledsoft decision parallel interference cancellation, used in the thirdgeneration of mobile telecommunication CDMA, is able to considerablyincrease the number of users such a system can host. We alsoconsider a variant of the well-known Hopfield model of neuralnetworks. We show that this variant proposed by Amari and Yanai[CITE] has a larger storage capacity than the original model. Both situations lead to the question of the moderatedeviations behavior of a sum of weakly dependent Bernoulli randomvariables. We prove a moderate deviations principle for such a sumon the appropriate scale.

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.

This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators,where we may use pattern functions or wavelets, and penalized maximumlikelihood estimators. In all these cases, we get oracle inequalities. In theformer we also have a polynomial rate of convergence for the non-parametricproblem. We finish the paper with applications of similar ideas to thecalibration of a photocounter, a measurement apparatus counting the number ofphotons in a beam. Here the mathematical problem reduces similarly to anon-parametric missingdata problem. We again get oracle inequalities, and better speed if thephotocounter is good.

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.

On donne des exemples d'entrelacements entre semi-groupes markoviens obtenus au moyen de considérations de théorie des groupes sur les paires de Gelfand

We present here a new proof of the theorem ofBirman and Solomyak on the metric entropy of the unit ball of aBesov space $B^s_{\pi,q}$ on a regular domain of ${\mathbb R}^d.$ Theresult is: if s - d(1/π - 1/p) +> 0, then the Kolmogorov metric entropy satisfies H(ε) ~ ε-d/s . This prooftakes advantage of the representation of such spaces on wavelet typebases and extends the result to more general spaces. The lower boundis a consequence of very simple probabilistic exponentialinequalities. To prove the upper bound, we provide a newuniversal coding based on a thresholding-quantizing procedure usingreplication.

To validate pollution data, subject-matter experts in Airpl (an organizationthat maintains a network of air pollution monitoring stations in westernFrance) daily perform visual examinations of the data and check theirconsistency. In this paper, we describe these visual examinations andpropose a formalization for this problem. The examinations consistin comparisons of so-called shorth intervals so we build a statisticaltest that compares such intervals in a nonparametric regression model.This allows to detect atypical data. A practical applicationof the test is given.

The aim is to study the asymptotic behavior of estimators and testsfor the components of identifiable finite mixture models ofnonparametric densities with a known number of components.Conditions for identifiability of the mixture components andconvergence of identifiable parameters are given.The consistency and weak convergence of the identifiable parametersand test statistics are presented for several models.