Deviation inequalities,moderate deviation principles for certain Gaussian functionals,and their applications in parameter estimation

In this article, we study the deviation inequalities, moderate deviation principle (MDP) and Berry–Esseen bounds for certain Gaussian functionals arising from the Ornstein-Uhlenbeck process without tears. As an application, several asymptotic properties for the minimum distance estimator are obtained. The main methods include the MDP and deviation inequality for multiple Wiener–Itô integrals.     

Moderate deviation principle for autoregressive processes

A moderate deviation principle for autoregressive processes is established. As statistical applications we provide the moderate deviation estimates of the least square and the Yule–Walker estimators of the parameter of an autoregressive process. The main assumption on the autoregressive process is the Gaussian integrability condition for the noise, which is weaker than the assumption of Logarithmic Sobolev Inequality in [H. Djellout, A. Guillin, L. Wu, Moderate deviations of empirical periodogram and nonlinear functionals of moving average processes, Ann. I. H. Poincaré-PR 42 (2006) 393–416].     

Dimension Free and Infinite Variance Tail Estimates on Poisson Space

Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including Lévy’s stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented.      

Deviation inequalities and moderate deviations for estimators of parameters in TAR models

In this paper, we establish some deviation inequalities and the moderate deviation principles for the least squares estimators of the parameters in the threshold autoregressive model under the assumption that the noise random variable satisfies a logarithmic Sobolev inequality.     

混合型随机微分方程的传输不等式

该文探讨一类由Wiener过程和Hurst参数1/2＜H＜1分数布朗运动驱动的混合型随机微分方程.通过使用一些变换技巧和逼近方法,这类方程的强解在d2度量和一致度量d∞下的二次传输不等式被建立.     

Optimal rates for parameter estimation of stationary Gaussian processes

We study rates of convergence in central limit theorems for partial sums of polynomial functionals of general stationary and asymptotically stationary Gaussian sequences, using tools from analysis on Wiener space. In the quadratic case, thanks to newly developed optimal tools, we derive sharp results, i.e. upper and lower bounds of the same order, where the convergence rates are given explicitly in the Wasserstein distance via an analysis of the functionals’ absolute third moments. These results are tailored to the question of parameter estimation, which introduces a need to control variance convergence rates. We apply our result to study drift parameter estimation problems for some stochastic differential equations driven by fractional Brownian motion with fixed-time-step observations.     

Some covariance inequalities in Wiener space

In this work we give an account of some covariance inequalities in abstract Wiener space. An FKG inequality is obtained with positivity and monotonicity being defined in terms of a given cone in the underlying Cameron-Martin space. The last part is dedicated to convex and log-concave functionals, including a proof of the Gaussian conjecture for a particular class of log-concave Wiener functionals.     

分数布朗运动随机微分方程的MLE和Bayes估计的大偏差不等式

本文研究了分数布朗运动随机微分方程未知参数的极大似然估计和Bayes估计的偏差不等式.在一定的正则条件下.利用似然方法给出了这两个估计量的大偏差不等式.     

On the Bohr Radius for Two Classes of Holomorphic Functions

Using some multidimensional analogs of the inequalities of E. Landau and F. Wiener for the Taylor coefficients of special classes of holomorphic functions on Reinhardt domains we obtain some estimates for the Bohr radius.     

Dimension-Independent Estimates on the Densities of Wiener Functionals via the Log-Sobolev Inequality

Using the log-Sobolev inequality, we shall present in this note some estimates on the density of finite dimensional non-degenerate Wiener functionals which are independent on the dimension. We shall take the Gaussian measure as the reference measure, contrary to the customary choice of Lebesgue measure in the literature. As an application, we show that the limit in probability of a uniformly bounded sequence of non-degenerate Wiener functionals has a density with respect to the Gaussian measure.     

Strong deviation theorems for functionals of the nonnegative continuous random variables

By using the approach of Laplace transform, we establish the strong deviation theorems represented by inequalities and described by asymptotic average log-likelihood ratio for functionals of the nonnegative continuous random variables. As corollaries, we obtain several strong laws of large numbers for the random variable functionals. The results of this paper extend and improve the corresponding results of some current literatures.     

Asymptotic distribution of some quadratic functionals of linear stochastic evolution systems

A function space asymptotic distribution of quadratic functionals induced from an unknown system is obtained in terms of a multi-dimensional Wiener process where the control is a linear transformation of the state that depends smoothly on the unknown parameters. The result is easily specialized to the asymptotic distribution of the family of random variables formed as the upper limit of the integrals of the quadratic terms is varied.The result provides a measure of the dependence of such a quadratic functional on a family of strongly consistent estimates of the unknown parameters, and in some cases it provides an interesting contrast with the case of all known parameters. In this paper, it is shown that, for some linear stochastic evolution systems, there are special feedback control laws where the variance of the asymptotic normal distribution of the average costs is less for the control law based on the estimates of the parameters than for the control law based on the true parameter values. This phenomenon does not occur if the feedback control laws are optimal stationary controls.This research was supported by NSF Grants Nos. ECS-87-18026 and ECS-9113029.The author thanks Professor Alain Benssousan for his great hospitality in INRIA, where this paper was written, and Professors Tyrone Duncan, Pravin Varaiya, and the anonymous reviewer for their very useful comments.     

Riesz transform and integration by parts formulas for random variables

We use integration by parts formulas to give estimates for the L^p norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and Thalmaier (2006) [13]. As a consequence, we obtain regularity and estimates for the density of non-degenerated functionals on the Wiener space. We also give a semi-distance which characterizes the convergence to the boundary of the set of the strict positivity points for the density.     

Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities

 In this paper, we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous Poisson process. We study consequently the adaptive properties of penalized projection estimators. At first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants. The crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of Poisson processes which are analogous to Talagrand's inequalities for empirical processes. Received: 24 April 2001 / Revised version: 9 October 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60E15, 62G05, 62G07 Key words or phrases: Inhomogeneous Poisson process – Concentration inequalities – Model selection – Penalized projection estimator – Adaptive estimation     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

Second order Poincaré inequalities and CLTs on Wiener space

We prove infinite-dimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Stein's method and Malliavin calculus. We provide two applications: (i) to a new “second order” characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.     

A new affine invariant for polytopes and Schneider's projection problem

New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.

     

Generalized calculus and sdes with non regular drift

In this note we prove a precise asymptotic estimate for Laplace type functionals for a parabolic SPDE. We use a large deviation principle, the stochastic Taylor expansion, some exponential inequalities and support theorems for our stochastic partial differential equation     

Déviations modérées pour la moyennisation d'une EDS avec petite diffusion

We prove in this Note the moderate deviation principle (MDP) for the averaging principle of a stochastic differential equation (SDE) in a fast random environment, modelized by an exponentially ergodic Markov process independent of the Wiener process driving the SDE. The main tools will be the method of Puhalskii for exponential tightness and a MDP for inhomogeneous functionals of Markov processes established in [5].     

Asymptotic behaviors for functionals of random dynamical systems

In this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.