Bernard Bercu

Bernard Bercu
University of Bordeaux · College of Science and Technology

PhD

About

117
Publications
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1,613
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September 2006 - February 2015
Université Bordeaux-I
Position
  • Professor (Full)

Publications

Publications (117)
Preprint
The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We prove that the Gram matrix associated with the MERWS, properly normalized, converges almost surely to the produc...
Preprint
Stochastic optimization naturally appear in many application areas, including machine learning. Our goal is to go further in the analysis of the Stochastic Average Gradient Accelerated (SAGA) algorithm. To achieve this, we introduce a new $\lambda$-SAGA algorithm which interpolates between the Stochastic Gradient Descent ($\lambda=0$) and the SAGA...
Preprint
Full-text available
The aim of this paper is to improve the large deviation principle for the number of descents in a random permutation by establishing a sharp large deviation principle of any order. We shall also prove a sharp large deviation principle of any order for the major index in a random permutation.
Preprint
Tools from optimal transport (OT) theory have recently been used to define a notion of quantile function for directional data. In practice, regularization is mandatory for applications that require out-of-sample estimates. To this end, we introduce a regularized estimator built from entropic optimal transport, by extending the definition of the ent...
Preprint
Full-text available
Chatteerjee and Diaconis have recently shown the asymptotic normality for the joint distribution of the number of descents and inverse descents in a random permutation. A noteworthy point of their results is that the asymptotic variance of the normal distribution is diagonal, which means that the number of descents and inverse descents are asymptot...
Article
Full-text available
The goal of this paper is to go further in the analysis of the behavior of the number of descents in a random permutation. Via two different approaches relying on a suitable martingale decomposition or on the Irwin–Hall distribution, we prove that the number of descents satisfies a sharp large-deviation principle. A very precise concentration inequ...
Preprint
Full-text available
We propose center-outward superquantile and expected shortfall functions, with applications to multivariate risk measurements, extending the standard notion of value at risk and conditional value at risk from the real line to $\RR^d$. Our new concepts are built upon the recent definition of Monge-Kantorovich quantiles based on the theory of optimal...
Preprint
Full-text available
We introduce a new stochastic algorithm for solving entropic optimal transport (EOT) between two absolutely continuous probability measures $\mu$ and $\nu$. Our work is motivated by the specific setting of Monge-Kantorovich quantiles where the source measure $\mu$ is either the uniform distribution on the unit hypercube or the spherical uniform dis...
Article
We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show t...
Preprint
Full-text available
The goal of this paper is to go further in the analysis of the behavior of the number of descents in a random permutation. Via two different approaches relying on a suitable martingale decomposition or on the Irwin-Hall distribution, we prove that the number of descents satisfies a sharp large deviation principle. A very precise concentration inequ...
Article
Full-text available
The aim of this paper is to go further in 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 resu...
Article
Full-text available
The aim of this paper is to investigate the asymptotic behavior of the so-called elephant random walk with stops (ERWS). In contrast with the standard elephant random walk, the elephant is allowed to be lazy by staying on his own position. We prove that the number of ones of the ERWS, properly normalized, converges almost surely to a Mittag–Leffler...
Article
Full-text available
We introduce a new second order stochastic algorithm to estimate the entropically regularized optimal transport (OT) cost between two probability measures. The source measure can be arbitrary chosen, either absolutely continuous or discrete, whereas the target measure is assumed to be discrete. To solve the semi-dual formulation of such a regulariz...
Preprint
Full-text available
The aim of this paper is to investigate the asymptotic behavior of the so-called elephant random walk with stops (ERWS). In contrast with the standard elephant random walk, the elephant is allowed to be lazy by staying on his own position. We prove that the number of ones of the ERWS, properly normalized, converges almost surely to a Mittag-Leffler...
Preprint
We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show t...
Preprint
Full-text available
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...
Article
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 rel...
Preprint
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 rel...
Preprint
Full-text available
We introduce a new second order stochastic algorithm to estimate the entropically regularized optimal transport cost between two probability measures. The source measure can be arbitrary chosen, either absolutely continuous or discrete, while the target measure is assumed to be discrete. To solve the semi-dual formulation of such a regularized and...
Article
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 quadratic strong law for the center...
Preprint
Full-text available
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 f...
Preprint
Full-text available
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 cente...
Article
A probabilistic approach is provided to establish new hypergeometric identities. It is based on the calculation of moments of the limiting distribution of the position of the elephant random walk in the superdiffusive regime.
Article
Full-text available
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 pap...
Article
The goal of this paper is to highlight the almost sure central limit theorem (ASCLT) for martingales to the control community. We shall establish the ASCLT for the least squares estimator of the unknown parameter of a controllable ARX(p,q) process in adaptive tracking. The usual notion of controllability for ARX(p,q) processes allows us to avoid th...
Preprint
Full-text available
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 arriv...
Article
Full-text available
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 normalit...
Preprint
A probabilistic approach is provided to establish new hypergeometric identities. It is based on the calculation of moments of the limiting distribution of the position of the elephant random walk in the superdiffusive regime.
Preprint
Full-text available
This paper is devoted to the stochastic approximation of entropically regularized Wasserstein distances between two probability measures, also known as Sinkhorn divergences. The semi-dual formulation of such regularized optimal transportation problems can be rewritten as a non-strongly concave optimisation problem. It allows to implement a Robbins-...
Preprint
Full-text available
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...
Preprint
Full-text available
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 autoreg...
Article
Full-text available
This paper is devoted to the estimation of the derivative of the regression function in fixed-design nonparametric regression. We establish the almost sure convergence as well as the asymptotic normality of our estimate. We also provide concentration inequalities which are useful for small sample sizes. Numerical experiments on simulated data show...
Data
Abstract: We study the rotation-activity correlations (RACs) in a sample of stars from spectral type dK4 to dM4. We study RACs using chromospheric data and coronal data. We study the Ca II line surface fluxes-P/sini RACs. We fit the RACs with linear homoscedastic and heteroscedastic regression models. We find that these RACs differ substantially fr...
Article
Full-text available
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 $0 \leq p <3/4$,...
Preprint
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 pap...
Preprint
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 $0 \leq p <3/4$,...
Article
Our goal is to establish large deviations 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 stabl...
Article
Full-text available
We study the rotation-activity correlations (RACs) in a sample stars from spectral type dK4 to dM4. We study RACs using chromospheric data and coronal data. We study the Ca\,{\sc ii} line surface fluxes-$P/\sin i$ RACs. We fit the RACs with linear homoscedastic and heteroscedastic regression models. We find that these RACs differ substantially from...
Preprint
We study the rotation-activity correlations (RACs) in a sample stars from spectral type dK4 to dM4. We study RACs using chromospheric data and coronal data. We study the Ca\,{\sc ii} line surface fluxes-$P/\sin i$ RACs. We fit the RACs with linear homoscedastic and heteroscedastic regression models. We find that these RACs differ substantially from...
Preprint
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 normalit...
Article
Full-text available
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....
Book
This chapter is devoted to concentration inequalities for martingales such as Azuma-Hoeffding, Freedman, and De la Pena inequalities. Several extensions will also be provided. In particular, we will focus our attention on improved versions of Azuma-Hoeffding and Freedman’s type inequalities.
Chapter
This chapter is devoted to standard asymptotic results for sums of independent random variables and martingales. Our goal is to provide a brief overview of basic results such as strong law of large numbers and central limit theorems, in order to set the stage for the rest of the book.
Chapter
This chapter is devoted to somme applications of concentration inequalities in probability and statistics. The first one deals with parameter estimation for autoregressive process and the second one is an improvement of a recent result on random permutations. The third one is a new result on empirical periodogram and the last one deals with random...
Article
The purpose of this book is to provide an overview of historical and recent results on concentration inequalities for sums of independent random variables and for martingales. The first chapter is devoted to classical asymptotic results in probability such as the strong law of large numbers and the central limit theorem. Our goal is to show that it...
Article
Full-text available
We investigate the asymptotic behaviour of the recursive 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...
Article
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 mo...
Article
Full-text available
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 Durbi...
Article
Full-text available
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...
Data
Full-text available
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 central limit theorem for martingales. A conjecture about almost sur...
Article
We investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein-Uhlenbeck processes driven by Ornstein-Uhlenbeck processes.
Article
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...
Article
We consider a semiparametric single index regression model involving a p-dimensional quantitative covariable x and a real dependent variable y. A dimension reduction is included in this model via an index x′β. Sliced inverse regression (SIR) is a well-known method to estimate the direction of the Euclidean parameter β which is based on a “slicing s...
Article
Full-text available
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 recursiv...
Article
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 theo...
Article
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...
Article
Full-text available
We investigate the spectral asymptotic properties of the stationary dynamical system $\xi_t=\varphi(T^t(X_0))$. This process is given by the iterations of a piecewise expanding map $T$ of the interval $[0,1]$, invariant for an ergodic probability $\mu$. The initial state $X_0$ is distributed over $[0,1]$ according to $\mu$ and $\varphi$ is a functi...
Article
Full-text available
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 sq...
Article
Full-text available
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...
Article
Full-text available
We obtain a large deviation principle for quadratic forms of Gaussian stationary processes. It is established by the conjunction of a result of S. Roch and B. Silbermann [Asymptotic Anal. 8, No. 3, 293–309 (1994; Zbl 0805.47024)] on the spectrum of products of Toeplitz matrices with the analysis of large deviations carried out by the first author,...
Article
We propose a new concept of strong controllability related to the Schur complement of a suitable limiting matrix. This new notion allows us to extend the previous convergence results associated with multidimensional ARX models in stochastic adaptive tracking. On the one hand, we carry out a sharp analysis of the almost sure convergence for both lea...
Article
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.
Conference Paper
Full-text available
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 l...
Preprint
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.
Article
Full-text available
We present a functional central limit theorem for a new class of interacting Markov chain Monte Carlo algorithms. These stochastic algorithms have been recently introduced to solve non-linear measure-valued equations. We provide an original theoretical analysis based on semigroup tech- niques on distribution spaces and fluctuation theorems for self...
Article
Full-text available
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...
Article
Full-text available
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 sum...
Article
Full-text available
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 l...
Article
Full-text available
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 central limit theorem for martingales. A conjecture about almost sur...
Article
Full-text available
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...
Conference Paper
Full-text available
We investigate the asymptotic properties of a recursive kernel density estimator associated with the driven noise of multivariate ARMAX models in adaptive tracking. We establish an almost sure pointwise and uniform strong law of large numbers as well as a pointwise and multivariate central limit theorem. We also carry out a goodness-of-fit test tog...
Article
Full-text available
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.
Article
We study the least-square (LS) estimator of the unknown parameters of a bifurcating auto-regressive process (BAR). Under very weak assumptions on the noise sequence (namely conditional pair-wise independence and moments of order $4$), we derive a precise rate of convergence for the LS estimator, as well as a quadratic strong law and a central limit...
Article
Full-text available
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 M...
Article
Full-text available
We study the least-square (LS) estimator of the unknown parameters of a bifurcating auto-regressive process (BAR). Under very weak assumptions on the noise sequence (namely conditional pair-wise independence and moments of order $4$), we derive a precise rate of convergence for the LS estimator, as well as a quadratic strong law and a central limit...
Article
Full-text available
We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments for powers of martingales. Several statistical applications on autoregressive and branching processes are also provided.
Preprint
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 algo...
Article
Full-text available
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 exa...
Conference Paper
Full-text available
We introduce a new concept of strong controllability for ARX models in adaptive tracking. This new notion is related to the Schur complement of a suitable limiting matrix. It allows us to extend the previous convergence results associated with both least squares and weighted least squares algorithms. In particular, we show the almost sure convergen...
Article
Full-text available
We investigate the asymptotic properties of a recursive kernel density estimator associated with the driven noise of a linear regression in adaptive tracking. We provide an almost sure pointwise and uniform strong law of large numbers as well as a pointwise and multivariate central limit theorem.
Article
Full-text available
We propose several exponential inequalities for self-normalized martingales similar to those established by De la Pena. 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.
Article
Full-text available
A wide range of literature concerning classical asymptotic properties for linear models with adaptive control is available, such as strong laws of large numbers or central limit theorems. Unfortunately, in contrast with the situation without control, it appears to be impossible to find sharp asymptotic or nonasymptotic properties such as large devi...
Book
Mathématiques appliquées pour le Master / SMAI
Article
Full-text available
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 empirica...
Article
Full-text available
In this report, new almost sure convergence properties for vectorial martingale transforms are established. Assuming some regularity conditions both on the increasing process and on the moments of the martingale, one can show that normalized moments of any even order do also converge in the almost sure cental limit theorem for martingales. Firm con...
Article
We establish new almost sure asymptotic properties for martingale transforms. It enables us to deduce the convergence of moments in the almost sure central limit theorem for martingales. Several statistical applications on the asymptotic behavior of stochastic regression models are also provided.
Article
The purpose of this note is to investigate the stability and the optimality of the adaptive tracking for a wide class of parametric nonlinear autoregressive models, via a new martingale approach. Several asymptotic results for the standard least squares estimator of the unknown model parameter, such as a central limit theorem, a law of iterated log...
Article
Full-text available
We consider the supremum $\mathcal{W}_n$ of self-normalized empirical processes indexed by unbounded classes of functions $\mathcal{F}$. Such variables are of interest in various statistical applications, for example, the likelihood ratio tests of contamination. Using the Herbst method, we prove an exponential concentration inequality for $\mathcal...
Article
We investigate the asymptotic behaviour of the empirical mode of an i.i.d. sample. We also study the asymptotic properties of the empirical spectral mode of a sample generated by a Gaussian stationary process. The empirical estimators are built using the empirical histogram or the empirical spectrogram, respectively. Under various regularity assump...

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