Bernard Bercu
University of Bordeaux
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Featured researches published by Bernard Bercu.
Stochastic Processes and their Applications | 1997
Bernard Bercu; Fabrice Gamboa; Alain Rouault
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.
Theory of Probability and Its Applications | 2002
Bernard Bercu; A. Rouault
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.
Siam Journal on Control and Optimization | 1995
Bernard Bercu
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.
Annals of Applied Probability | 2008
Bernard Bercu; Abderrahmen Touati
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.
Theory of Probability and Its Applications | 2011
Bernard Bercu; Laure Coutin; Nicolas Savy
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.
IEEE Transactions on Automatic Control | 2009
Bernard Bercu; François Dufour; G. Yin
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.
Siam Journal on Control and Optimization | 1998
Bernard Bercu
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.
Annals of Statistics | 2012
Bernard Bercu; Philippe Fraysse
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.
Bernoulli | 2001
Bernard Bercu
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.
Archive | 2015
Bernard Bercu; Bernard Delyon; Emmanuel Rio
Classical Results.- Concentration Inequalities for Sums.- Concentration Inequalities for Martingales.- Applications in Probability and Statistics.