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.

Let M be acomplete Riemannian manifold, M ∈ ℕ andp ≥ 1. Weprove that almost everywhere on x = (x1,...,xN) ∈ MNfor Lebesgue measure in MN, the measure \hbox{$\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$}$\mathit{\mu }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{N}}{\mathit{\delta }}_{{\mathit{x}}_{\mathit{k}}}$ has a unique p–mean ep(x).As a consequence, if X = (X1,...,XN)is a MN-valued randomvariable with absolutely continuous law, then almost surely μ(X(ω)) has aunique p–mean. In particular if (Xn)n ≥ 1is an independent sample of an absolutely continuous law in M, then the processep,n(ω) = ep(X1(ω),...,Xn(ω))is well-defined. Assume M is compact and consider a probability measureν inM. Usingpartial simulated annealing, we define a continuous semimartingale which converges inprobability to the set of minimizers of the integral of distance at power p with respect toν. When theset is a singleton, it converges to the p–mean.

We prove existence and uniqueness for two classes of martingale problems involving a nonlinear but bounded drift coefficient. In the first class, this coefficient depends on the time t, the position x and the marginal of the solution at time t. In the second, it depends on t, x and p(t,x), the density of the time marginal w.r.t. Lebesgue measure. As far as the dependence on t and x is concerned, no continuity assumption is made. The results, first proved for the identity diffusion matrix, are extended to bounded, uniformly elliptic and Lipschitz continuous matrices. As an application, we show that within each class, a particular choice of the coefficients leads to a probabilistic interpretation of generalizations of Burgers' equation.

We study the convergence to equilibrium of n-samples of independent Markovchains in discrete and continuous time. They are defined as Markov chains on the n-fold Cartesian product of the initial state space by itself, and they converge to the direct product of n copies of the initial stationary distribution. Sharp estimates for the convergence speed are given in terms of the spectrum of the initial chain. A cutoff phenomenon occurs in thesense that as n tends to infinity, the total variation distancebetween the distribution of the chain and the asymptotic distribution tendsto 1 or 0 at all times. As an application,an algorithm is proposed for producing an n-sample of the asymptotic distribution of the initial chain, with an explicit stopping test.

In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurablecoefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute thesequantities by solving some suitableelliptic PDE problems.

In this note we propose an exact simulation algorithm for the solution of (1)

\begin{equation} \label{eds-intro} {\rm d}X_t={\rm d}W_t+\bar{b}(X_t){\rm d}t,\quad X_0=x, \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}\mathrm{\right)}\mathrm{d}\mathit{t,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x,}$

\hbox{$\bar{b}$}b̅

\hbox{$\bar{b}(0+)\neq \bar{b}(0-)$}b̅(0 + ) ≠b̅(0 −)

(2)

\begin{equation} \label{edsbeta} {\rm d}X^\beta_t={\rm d}W_t+\bar{b}(X^\beta_t){\rm d}t + \beta {\rm d}L^0_t(X^\beta),\quad X_0=x \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{\right)}\mathrm{d}\mathit{t}\mathrm{+}\mathit{\beta }\mathrm{d}{\mathit{L}}_{\mathit{t}}^{\mathrm{0}}\mathrm{\left(}{\mathit{X}}^{\mathit{\beta }}\mathrm{\right)}\mathit{,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x}$

In Francq and Zakoïan [4], we derived stationarity conditions forARMA(p,q) models subject to Markov switching. In this paper, weshow that, under appropriate moment conditions, the powers of thestationary solutions admit weak ARMA representations, which we areable to characterize in terms of p,q, the coefficients of themodel in each regime, and the transition probabilities of theMarkov chain. These representations are potentially useful forstatistical applications.

We study strictly parabolic stochastic partial differential equations on $\mathbb{R}^d$ , d ≥ 1,driven by a Gaussian noise white in time and coloured in space. Assuming that thecoefficients of the differential operator are random, we give sufficient conditions on thecorrelation of the noise ensuring Hölder continuity for the trajectories of thesolution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weakformulation of the equation are related, deriving path properties of the solution to aparabolic Cauchy problem in evolution form.


This paper is dedicated to the analysis of backward stochastic differential equations(BSDEs) with jumps, subject to an additional global constraint involving all thecomponents of the solution. We study the existence and uniqueness of a minimal solutionfor these so-called constrained BSDEs with jumps via a penalizationprocedure. This new type of BSDE offers a nice and practical unifying framework to thenotions of constrained BSDEs presented in [S. Peng and M. Xu, Preprint.(2007)] and BSDEs with constrained jumps introduced in [I. Kharroubi, J. Ma, H.Pham and J. Zhang, Ann. Probab. 38 (2008) 794–840]. Moreremarkably, the solution of a multidimensional Brownian reflected BSDE studied in [Y. Huand S. Tang, Probab. Theory Relat. Fields 147 (2010) 89–121]and [S. Hamadène and J. Zhang, Stoch. Proc. Appl. 120 (2010)403–426] can also be represented via a well chosen one-dimensionalconstrained BSDE with jumps. This last result is very promising from a numerical point ofview for the resolution of high dimensional optimal switching problems and more generallyfor systems of coupled variational inequalities.

Let \hbox{$\mathbb{F}^{}_{}$}$F_{}^{}$ be a filtration and τ be a random time. Let \hbox{$\mathbb{G}^{}_{}$}$G_{}^{}$ be the progressive enlargement of \hbox{$\mathbb{F}^{}_{}$}$F_{}^{}$ with τ. We study the following formula, called the optional splitting formula: For any \hbox{$\mathbb{G}^{}_{}$}$G_{}^{}$-optional process Y, there exists an \hbox{$\mathbb{F}^{}_{}$}$F_{}^{}$-optional process Y′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being \hbox{$\mathcal{B}[0,\infty]\otimes\mathcal{O}(\mathbb{F}^{}_{})$}$\mathrm{ℬ}\mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{\infty }\mathrm{\right]}\mathrm{\otimes }\mathrm{𝒪}\mathrm{\left(}{F}_{}^{}\mathrm{\right)}$ measurable, such that \hbox{$Y=Y'\mathds{1}^{}_{[0,\tau)}+Y''(\tau)\mathbb{1}^{}_{[\tau,\infty)}.$}$\mathit{Y}\mathrm{=}{\mathit{Y}}^{\mathrm{\prime }}{1}_{\mathrm{\left[}\mathrm{0}\mathit{,\tau }\mathrm{\right)}}^{}\mathrm{+}{\mathit{Y}}^{\mathrm{\prime \prime }}\mathrm{\left(}\mathit{\tau }\mathrm{\right)}{1}_{\mathrm{\left[}\mathit{\tau ,}\mathrm{\infty }\mathrm{\right)}}^{}\mathit{.}$ (This formula can also be formulated for multiple random times τ1,...,τk). We are interested in this formula because of its fundamental role in many recent papers on credit risk modeling, and also because of the fact that its validity is limited in scope and this limitation is not sufficiently underlined. In this paper we will determine the circumstances in which the optional splitting formula is valid. We will then develop practical sufficient conditions for that validity. Incidentally, our results reveal a close relationship between the optional splitting formula and several measurability questions encountered in credit risk modeling. That relationship allows us to provide simple answers to these questions.

We establish optimal uniform upper estimates on heat kernels whose generators satisfy a logarithmic Sobolev inequality (or entropy-energy inequality) with the optimal constant of the Euclidean space. Off-diagonals estimates may also be obtained with however a smaller distance involving harmonic functions. In the last part, we apply these methods to study some heat kernel decays for diffusion operators of the type Laplacian minus the gradient of a smooth potential with a given growth at infinity.

Sample path large deviationsfor the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero arepresented. The noise is a complex additive Gaussian noise. It iswhite in time and colored in space. The solutions may be global orblow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologiesanalogue to projective limit topologies. In this setting, thesupport of the law of the solution is also characterized. As aconsequence, results on the law of the blow-up time andasymptotics when the noise converges to zero are obtained. Anapplication to the transmission of solitary waves in fiber opticsis also given.

The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin–Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. In addition, the almost sure rates of convergence of our estimates are also provided. It allows us to establish the almost sure convergence and the asymptotic normality for the Durbin–Watson statistic. Finally, we propose a new bilateral statistical test for residual autocorrelation. We show how our statistical test procedure performs better, from a theoretical and a practical point of view, than the commonly used Box–Pierce and Ljung–Box procedures, even on small-sized samples.

We solve explicitly the following problem: for a given probability measure μ, we specify a generalised martingale diffusion (Xt) which, stopped at an independent exponential time T, is distributed according to μ. The process (Xt) is specified via its speed measure m. We present two heuristic arguments and three proofs. First we show how the result can be derived from the solution of [Bertoin and Le Jan, Ann. Probab. 20 (1992) 538–548.] to the Skorokhod embedding problem. Secondly, we give a proof exploiting applications of Krein's spectral theory of strings to the study of linear diffusions. Finally, we present a novel direct probabilistic proof based on a coupling argument.

(Local) self-similarity is a seminal concept, especially for Euclidean random fields. Westudy in this paper the extension of these notions to manifold indexed fields. We giveconditions on the (local) self-similarity index that ensure the existence of fractionalfields. Moreover, we explain how to identify the self-similar index. We describe a way ofsimulating Gaussian fractional fields.

Aggregation patterns are often visually detected in sets of location data. These clustersmay be the result of interesting dynamics or the effect of pure randomness. We build anasymptotically Gaussian test for the hypothesis of randomness corresponding to ahomogeneous Poisson point process. We first compute the exact first and second moment ofthe Ripley K-statistic under the homogeneous Poisson point process model.Then we prove the asymptotic normality of a vector of such statistics for different scalesand compute its covariance matrix. From these results, we derive a test statistic that ischi-square distributed. By a Monte-Carlo study, we check that the test is numericallytractable even for large data sets and also correct when only a hundred of points areobserved.

In this paper we consider three measures of overlap, namely Matusia's measure ρ, Morisita's measure λ andWeitzman's measure Δ. These measures are usually used inquantitative ecology and stress-strength models of reliabilityanalysis. Herein we consider two Weibull distributions havingthe same shape parameter and different scale parameters. Thisdistribution is known to be the most flexible life distributionmodel with two parameters. Monte Carlo evaluations are used tostudy the bias and precision of some estimators of these overlapmeasures. Confidence intervals for the measures are alsoconstructed via bootstrap methods and Taylor series approximation.

Gaussian mixture models are widely used to study clustering problems. These model-based clustering methods require an accurate estimation of the unknown data density by Gaussian mixtures. In Maugis and Michel (2009), a penalized maximum likelihood estimator is proposed for automatically selecting the number of mixture components. In the present paper, a collection of univariate densities whose logarithm is locally β-Hölder with moment and tail conditions are considered. We show that this penalized estimator is minimax adaptive to the β regularity of such densities in the Hellinger sense.

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.


This paper deals with the problem of estimating a regression function f,in a random design framework. We build and study two adaptive estimators based on modelselection, applied with warped bases. We start with a collection of finite dimensionallinear spaces, spanned by orthonormal bases. Instead of expanding directly the targetfunction f on these bases, we rather consider the expansion ofh = f ∘ G-1, whereG is the cumulative distribution function of the design, followingKerkyacharian and Picard [Bernoulli 10 (2004) 1053–1105].The data-driven selection of the (best) space is done with two strategies: we use both apenalization version of a “warped contrast”, and a model selection device in the spirit ofGoldenshluger and Lepski [Ann. Stat. 39 (2011) 1608–1632].We propose by these methods two functions, ĥl(l = 1, 2), easier to compute than least-squares estimators. Weestablish nonasymptotic mean-squared integrated risk bounds for the resulting estimators, \hbox{$\hat{f}_l=\hat{h}_l\circ G$}f̂l = ĥl°G if G is known, or\hbox{$\hat{f}_l=\hat{h}_l\circ\hat{G}$}f̂l = ĥl°Ĝ(l = 1,2) otherwise, where Ĝ is theempirical distribution function. We study also adaptive properties, in case the regressionfunction belongs to a Besov or Sobolev space, and compare the theoretical and practicalperformances of the two selection rules.