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Bernard Bercu

Publications and source records attributed to Bernard Bercu.

At least 37 records · Page 2Linked to original sources
arXiv2021

New insights on the minimal random walk

The aim of this paper is to deepen the analysis of the asymptotic behavior of the so-called minimal random walk (MRW) using a new martingale approach. The MRW is a discrete-time random walk with infinite memory that has three regimes depending on the location of its two parameters. In the diffusive and critical regimes, we establish new results on the almost sure asymptotic behavior of the MRW, such as the quadratic strong law and the law of the iterated logarithm. In the superdiffusive regime, we prove the almost sure convergence of the MRW, properly normalized, to a nondegenerate random variable. Moreover, we show that the fluctuation of the MRW around its limiting random variable is still Gaussian.

math.PR↗
arXiv2021

Asymptotic analysis of random walks on ice and graphite

The purpose of this paper is to investigate the asymptotic behavior of random walks on three-dimensional crystal structures. We focus our attention on the 1h structure of the ice and the 2h structure of graphite. We establish the strong law of large numbers and the asymptotic normality for both random walks on ice and graphite. All our analysis relies on asymptotic results for multi-dimensional martingales.

math-ph↗
arXiv2020

Stochastic approximation algorithms for superquantiles estimation

This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.

math.ST↗
arXiv2020

On the center of mass of the elephant random walk

Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratric strong law for the center of mass of the elephant random walk. The asymptotic normality of the center of mass, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales.

math.PR↗
arXiv2019

New insights on concentration inequalities for self-normalized martingales

We propose new concentration inequalities for self-normalized martingales. The main idea is to introduce a suitable weighted sum of the predictable quadratic variation and the total quadratic variation of the martingale. It offers much more flexibility and allows us to improve previous concentration inequalities. Statistical applications on autoregressive process, internal diffusion-limited aggregation process, and online statistical learning are also provided.

math.PR↗
arXiv2019

An efficient stochastic Newton algorithm for parameter estimation in logistic regressions

Logistic regression is a well-known statistical model which is commonly used in the situation where the output is a binary random variable. It has a wide range of applications including machine learning, public health, social sciences, ecology and econometry. In order to estimate the unknown parameters of logistic regression with data streams arriving sequentially and at high speed, we focus our attention on a recursive stochastic algorithm. More precisely, we investigate the asymptotic behavior of a new stochastic Newton algorithm. It enables to easily update the estimates when the data arrive sequentially and to have research steps in all directions. We establish the almost sure convergence of our stochastic Newton algorithm as well as its asymptotic normality. All our theoretical results are illustrated by numerical experiments.

math.ST↗
arXiv2018

On the almost sure central limit theorem for ARX processes in adaptive tracking

The goal of this paper is to highlight the almost sure central limit theorem for martingales to the control community and to show the usefulness of this result for the system identification of controllable ARX(p,q) process in adaptive tracking. We also provide strongly consistent estimators of the even moments of the driven noise of a controllable ARX(p,q) process as well as quadratic strong laws for the average costs and estimation errors sequences. Our theoretical results are illustrated by numerical experiments.

math.OC↗
arXiv2017

On the multi-dimensional elephant random walk

The purpose of this paper is to investigate the asymptotic behavior of the multi-dimensional elephant random walk (MERW). It is a non-Markovian random walk which has a complete memory of its entire history. A wide range of literature is available on the one-dimensional ERW. Surprisingly, no references are available on the MERW. The goal of this paper is to fill the gap by extending the results on the one-dimensional ERW to the MERW. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the MERW. The asymptotic normality of the MERW, properly normalized, is also provided. In the superdiffusive regime, we prove the almost sure convergence as well as the mean square convergence of the MERW. All our analysis relies on asymptotic results for multi-dimensional martingales.

math.PR↗
arXiv2017

A martingale approach for the elephant random walk

The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter $p$ which lies between zero and one. This behavior is totally different in the diffusive regime $0 \leq p <3/4$, the critical regime $p=3/4$, and the superdiffusive regime $3/4<p \leq 1$. Notwithstanding of this trichotomy, we provide some new results on the almost sure convergence and the asymptotic normality of the ERW.

math.PR↗
arXiv2016

Nonparametric estimation of the derivative of the regression function: application to sea shores water quality

This paper is devoted to the nonparametric estimation of the derivative of the regression function in a nonparametric regression model. We implement a very efficient and easy to handle statistical procedure based on the derivative of the recursive Nadaraya-Watson estimator. We establish the almost sure convergence as well as the asymptotic normality for our estimates. We also illustrate our nonparametric estimation procedure on simulated and real life data associated with sea shores water quality and valvometry.

math.ST↗
arXiv2016

Large deviations and concentration inequalities for the Ornstein-Uhlenbeck process without tears

Our goal is to establish large deviations and concentration inequalities for the maximum likelihood estimator of the drift parameter of the Ornstein-Uhlenbeck process without tears. We propose a new strategy to establish large deviation results which allows us, via a suitable transformation, to circumvent the classical difficulty of non-steepness. Our approach holds in the stable case where the process is positive recurrent as well as in the unstable and explosive cases where the process is respectively null recurrent and transient. Notwithstanding of this trichotomy, we also provide new concentration inequalities for the maximum likelihood estimator.

math.ST↗
arXiv2014

Large deviations for the Ornstein-Uhlenbeck process with shift

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We estimate simultaneously the drift and shift parameters. On the one hand, we establish a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the one previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

math.PR↗
arXiv2014

A Durbin-Watson serial correlation test for ARX processes via excited adaptive tracking

We propose a new statistical test for the residual autocorrelation in ARX adaptive tracking. The introduction of a persistent excitation in the adaptive tracking control allows us to build a bilateral statistical test based on the well-known Durbin-Watson statistic. We establish the almost sure convergence and the asymptotic normality for the Durbin-Watson statistic leading to a powerful serial correlation test. Numerical experiments illustrate the good performances of our statistical test procedure.

math.PR↗
arXiv2012

A Rademacher-Menchov approach for random coefficient bifurcating autoregressive processes

We investigate the asymptotic behavior of the least squares estimator of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on inherited and environmental effects, we establish the almost sure convergence of our estimates. In addition, we also prove a quadratic strong law and central limit theorems. Our approach mainly relies on asymptotic results for vector-valued martingales together with the well-known Rademacher-Menchov theorem.

math.PR↗
arXiv2012

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 very efficient and easy to handle Robbins-Monro procedure. 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 to estimate 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 into 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. Finally, we illustrate our semiparametric estimation procedure on simulated and real data.

math.ST↗
arXiv2012

On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking

A wide literature is available on the asymptotic behavior of the Durbin-Watson statistic for autoregressive models. However, it is impossible to find results on the Durbin-Watson statistic for autoregressive models with adaptive control. Our purpose is to fill the gap by establishing the asymptotic behavior of the Durbin Watson statistic for ARX models in adaptive tracking. On the one hand, we show the almost sure convergence as well as the asymptotic normality of the least squares estimators of the unknown parameters of the ARX models. On the other hand, we establish the almost sure convergence of the Durbin-Watson statistic and its asymptotic normality. Finally, we propose a bilateral statistical test for residual autocorrelation in adaptive tracking.

math.ST↗