Let θ be the mode of a probability density and θn itskernel estimator. In the case θ is nondegenerate, we first specify the weakconvergence rate of the multivariate kernel mode estimator by stating the central limittheorem for θn - θ. Then, we obtain a multivariate law ofthe iterated logarithm for the kernel mode estimator by proving that, with probabilityone, the limit set of the sequence θn - θ suitably normalized is an ellipsoid.We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], ofθn - θ. Finally, we consider the case θ is degenerate and give the exactweak and strong convergence rate of θn - θ in the univariate framework.

In this paper a new multifractal stochastic process called Limit of theIntegrated Superposition of Diffusion processes with Linear differencialGenerator (LISDLG) is presented which realistically characterizes the networktraffic multifractality. Several properties of the LISDLG model are presentedincluding long range dependence, cumulants, logarithm of the characteristicfunction, dilative stability, spectrum and bispectrum. The model captureshigher-order statistics by the cumulants. The relevance and validation of theproposed model are demonstrated by real data of Internet traffic.


We consider the random vector $u(t,\underlinex)=(u(t,x_1),\dots,u(t,x_d))$ , where t > 0, x1,...,xd aredistinct points of $\mathbb{R}^2$ and u denotes the stochastic process solution to a stochastic waveequation driven bya noise white in time and correlated in space. In a recent paper byMillet and Sanz–Solé[10], sufficient conditions are given ensuring existence andsmoothness ofdensity for $u(t,\underline x)$ . We study here the positivity of suchdensity. Usingtechniques developped in [1] (see also [9]) basedon Analysis on anabstract Wiener space, we characterize the set of points $y\in\mathbb{R}^d$ where the density ispositive and we prove that, under suitable assumptions, this set is $\mathbb{R}^d$ .

We present a spectral theory for a class of operators satisfying a weak“Doeblin–Fortet" condition and apply it to a class of transition operators.This gives the convergence of the series ∑k≥0krPkƒ, $r \in \mathbb{N}$ ,under some regularity assumptions and implies the central limit theoremwith a rate in $n^{- \frac{1}{2} }$ for the corresponding Markov chain.An application to a non uniformly hyperbolic transformation on the interval is also given.

We propose a test of a qualitative hypothesis on the mean of a n-Gaussianvector. The testing procedure is available when the variance of theobservations is unknown and does not depend on any prior information onthe alternative. The properties of the test are non-asymptotic. Fortesting positivity or monotonicity, weestablish separation rates with respect to the Euclidean distance, oversubsets of $\mathbb{R}^{n}$ which are related to Hölderian balls in functionalspaces. We provide a simulation study in order to evaluate theprocedure when the purpose is to test monotonicity in a functionalregression model and to check the robustness of the procedure tonon-Gaussian errors.

In this paper we solve the basic fractionalanalogue of the classical linear-quadratic Gaussianregulator problem in continuous time. For a completelyobservable controlled linear system driven by a fractionalBrownian motion, we describe explicitely the optimal controlpolicy which minimizes a quadratic performance criterion.

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue ofSchrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine a class of modelsextending the hard obstacle modelof K. Burdzy, R. Holyst and P. March and including the Moran type scheme presented by the authors in a previous work. We provide precise uniform estimates with respect to the time parameter and we analyze the fluctuations of continuous time particle models.

We derive necessary and sufficient conditions for a sum of i.i.d.random variables $\sum_{i=1}^n X_i/b_n$ –where $\frac {b_n} n \downarrow 0$ ,but $\frac {b_n} {\sqrt n} \uparrow \infty$ – to satisfy a moderate deviationsprinciple. Moreover we show that this equivalence is a typical moderatedeviations phenomenon. It is not true in a large deviations regime.

The P.O.T. (Peaks-Over-Threshold) approachconsists of using the Generalized ParetoDistribution (GPD)to approximate the distribution of excesses over a threshold.We use the probability-weighted momentsto estimate the parameters of the approximating distribution.We study the asymptotic behaviour ofthese estimators (in particular their asymptotic bias) and also thefunctional bias of the GPD as an estimate of thedistribution function of the excesses. We adapt penultimateapproximation results to the case where parameters are estimated.

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.

In this paper, we prove that the laws of interacting Brownian particlesare characterized as Gibbs fields on pathspace associated to an explicit class ofHamiltonian functionals. More generally, we show that a large class of Gibbsfields on pathspace corresponds to Brownian diffusions. Some applications totime reversal in the stationary and non stationary case are presented.

In the problem of signal detectionin Gaussian white noisewe show asymptotic minimaxity of kernel-based tests. The test statisticsequal L 2-norms of kernel estimates.The sets of alternatives are essentially nonparametric and are defined asthe sets of all signals such that the L 2-norms of signal smoothed by the kernels exceed some constants pε > 0.The constant pε depends on the power ϵof noise and pε → 0 as ε → 0.Similar statements are proved also if an additional informationon a signal smoothness is given.By theorems on asymptotic equivalence of statistical experimentsthese results are extended to the problems of testing nonparametrichypotheseson density and regression. The exact asymptotically minimaxlower bounds of type II error probabilities are pointed out forall these settings. Similar results are also obtained for the problemsof testing parametric hypotheses versus nonparametric sets of alternatives.

Let (X1,Y1),...,(Xm,Ym ) be m independent identicallydistributed bivariate vectorsandL1 = β1X1 + ... + βmXm , L2 = β1X1 + ... + βmXm are two linear forms with positive coefficients.We study two problems:under what conditions does the equidistribution of L 1 and L 2imply the same property forX 1 and Y 1, and under what conditions does the independence of L 1and L 2 entail independenceof X 1 and Y 1?Some analytical sufficient conditions are obtained and it is shownthat in general they can not be weakened.