Bernard Bercu

Large deviations for quadratic forms of stationary Gaussian processes

A large deviation principle is proved for Toeplitz quadratic forms of centred stationary Gaussian processes. The rate function is obtained by a sharp study of the behaviour of eigenvalues of a product of two Toeplitz matrices. Some statistical applications such as the likelihood ratio test and the estimation of the parameter of an autoregressive Gaussian process are also provided.

Sharp Large Deviations for the Ornstein--Uhlenbeck Process

We establish sharp large deviation principles for well-known random variables associated with the Ornstein--Uhlenbeck process, such as the energy, the maximum likelihood estimator of the drift parameter, and the log-likelihood ratio.

Weighted Estimation and Tracking for ARMAX Models

For complex multivariate ARMAX models, the author studies the weighted least squares algorithm which offers, by the choice of suitable weightings, the advantages of both the extended least squares and the stochastic gradient algorithms. Concerning adaptive tracking problems, the strong consistency of the estimator and control optimality are both ensured. Almost sure rates of convergence are also provided.

Exponential inequalities for self-normalized martingales with applications

We propose several exponential inequalities for self-normalized martingales similar to those established by De la Pe\~{n}a. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided.

Sharp large deviations for the fractional Ornstein-Uhlenbeck process

We investigate the sharp large deviation properties of the energy and the maximum likelihood estimator for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst index greater than one half.

Almost Sure Stabilization for Feedback Controls of Regime-Switching Linear Systems With a Hidden Markov Chain

This work is devoted to the almost sure stabilization of adaptive control systems that involve an unknown Markov chain. The control system displays continuous dynamics represented by differential equations and discrete events given by a hidden Markov chain. In the previous investigation on this class of problems, averaging criteria were used, which provides only the system behavior in some expectation sense. A closer scrutiny of the system behavior necessarily requires the consideration of sample path properties. Different from previous work on stabilization of adaptive controlled systems with a hidden Markov chain, where average criteria were considered, this work focuses on the almost sure stabilization or sample path stabilization of the underlying processes. Under simple conditions, it is shown that as long as the feedback controls have linear growth in the continuous component, the resulting process is regular. Moreover, by appropriate choice of the Lyapunov functions, it is shown that the adaptive system is stabilizable almost surely. As a by-product, it is also established that the controlled process is positive recurrent.

Central Limit Theorem And Law Of Iterated Logarithm For Least Squares Algorithms In Adaptive Tracking

In autoregressive adaptive tracking, we prove that the least squares and the weighted least squares algorithms possess the same asymptotic properties, sharing the same central limit theorem and the same law of iterated logarithm. We also obtain the same asymptotic behavior and show the limitations of these results in the autoregressive with moving average framework.

A Robbins-Monro procedure for estimation in semiparametric regression models

This paper is devoted to the parametric estimation of a shift together with the nonparametric estimation of a regression function in a semiparametric regression model. We implement a Robbins-Monro procedure very efficient and easy to handle. On the one hand, we propose a stochastic algorithm similar to that of Robbins-Monro in order to estimate the shift parameter. A preliminary evaluation of the regression function is not necessary for estimating the shift parameter. On the other hand, we make use of a recursive Nadaraya-Watson estimator for the estimation of the regression function. This kernel estimator takes in account the previous estimation of the shift parameter. We establish the almost sure convergence for both Robbins-Monro and Nadaraya-Watson estimators. The asymptotic normality of our estimates is also provided.

On large deviations in the Gaussian autoregressive process: stable, unstable and explosive cases

For the Gaussian autoregressive process, the asymptotic behaviour of the Yule‐Walker estimator is totally different in the stable, unstable and explosive cases. We show that, irrespective of this trichotomy, this estimator shares quite similar large deviation properties in the three situations. However, in the explosive case, we obtain an unusual rate function with a discontinuity point at its minimum.

Concentration Inequalities for Sums and Martingales

Classical Results.- Concentration Inequalities for Sums.- Concentration Inequalities for Martingales.- Applications in Probability and Statistics.