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Bernard Bercu

Publications and source records attributed to Bernard Bercu.

51 records · Page 3Linked to original sources
arXiv2012

On the asymptotic behavior of the Nadaraya-Watson estimator associated with the recursive SIR method

We investigate the asymptotic behavior of the Nadaraya-Watson estimator for the estimation of the regression function in a semiparametric regression model. On the one hand, we make use of the recursive version of the sliced inverse regression method for the estimation of the unknown parameter of the model. On the other hand, we implement a recursive Nadaraya-Watson procedure for the estimation of the regression function which takes into account the previous estimation of the parameter of the semiparametric regression model. We establish the almost sure convergence as well as the asymptotic normality for our Nadaraya-Watson estimator. We also illustrate our semiparametric estimation procedure on simulated data.

math.ST↗
arXiv2012

Fluctuations of Interacting Markov Chain Monte Carlo Methods

We present a multivariate central limit theorem for a general class of interacting Markov chain Monte Carlo algorithms used to solve nonlinear measure-valued equations. These algorithms generate stochastic processes which belong to the class of nonlinear Markov chains interacting with their empirical occupation measures. We develop an original theoretical analysis based on resolvent operators and semigroup techniques to analyze the fluctuations of their occupation measures around their limiting values.

math.PR↗
arXiv2011

Sharp large deviations for the non-stationary Ornstein-Uhlenbeck process

For the Ornstein-Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding of this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.

math.PR↗
arXiv2011

A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process

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 to 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.

math.ST↗
arXiv2009

Almost sure central limit theorems on the Wiener space

In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law to a normal distribution.

math.PR↗
arXiv2009

Asymptotic analysis for bifurcating autoregressive processes via a martingale approach

We study the asymptotic behavior of the least squares estimators of the unknown parameters of bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the central limit theorem. All our analysis relies on non-standard asymptotic results for martingales.

math.PR↗
arXiv2009

A multiple stochastic integral criterion for almost sure limit theorems

In this paper, we study almost sure central limit theorems for multiple stochastic integrals and provide a criterion based on the kernel of these multiple integrals. We apply our result to normalized partial sums of Hermite polynomials of increments of fractional Brownian motion. We obtain almost sure central limit theorems for these normalized sums when they converge in law to a normal distribution.

math.PR↗
arXiv2009

On the usefulness of persistent excitation in ARX adaptive tracking

The usefulness of persistent excitation is well-known in the control community. Thanks to a persistently excited adaptive tracking control, we show that it is possible to avoid the strong controllability assumption recently proposed in the multidimensional ARX framework. We establish the almost sure convergence for both least squares and weighted least squares estimators of the unknown parameters. A central limit theorem and a law of iterated logarithm are also provided. All this asymptotical analysis is related to the Schur complement of a suitable limiting matrix.

math.PR↗
arXiv2008

On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure cental limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

math.PR↗
arXiv2008

Exponential inequalities for self-normalized martingales with applications

We propose several exponential inequalities for self-normalized martingales similar to those established by De la Peña. 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.

math.ST↗
arXiv2008

Almost Sure Stabilization for Adaptive Controls of Regime-switching LQ 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. 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.

math.PR↗
arXiv2008

A new concept of strong controllability via the Schur complement in adaptive tracking

We propose a new concept of strong controllability associated with the Schur complement of a suitable limiting matrix. This concept allows us to extend the previous results associated with multidimensional ARX models. On the one hand, we carry out a sharp analysis of the almost sure convergence for both least squares and weighted least squares algorithms. On the other hand, we also provide a central limit theorem and a law of iterated logarithm for these two stochastic algorithms. Our asymptotic results are illustrated by numerical simulations.

math.PR↗
arXiv2007

Spectrum of the product of Toeplitz matrices with application in probability

We study the spectrum of the product of two Toeplitz operators. Assume that the symbols of these operators are continuous and real-valued and that one of them is non-negative. We prove that the spectrum of the product of finite section Toeplitz matrices converges to the spectrum of the product of the semi-infinite Toeplitz operators. We give an example showing that the supremum of this set is not always the supremum of the product of the two symbols. Finally, we provide an application in probability which is the first motivation of this study. More precisely, we obtain a large deviation principle for Gaussian quadratic forms.

math.FA↗
arXiv2006

Asymptotic results for empirical measures of weighted sums of independent random variables

We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This implies almost sure convergence of empirical periodograms, almost sure convergence of spectra of circulant and reverse circulant matrices, and almost sure convergence of the CDF's generated from independent random variables by independent random orthogonal matrices. For special trigonometric matrices, the speed of the almost sure convergence is described by the normal approximation and by the large deviation principle.

math.PR↗