Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of... more Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of the well-known Fractional Brownian Motion to the framework of heavytailed stable distributions. It was introduced several decades ago; however its properties are far from being completely understood. In our present paper we determine the optimal power of the logarithmic factor in a uniform modulus of continuity for HFSM, which solves an open old problem. The keystone of our strategy consists in Abel transforms of the LePage series expansions of the random coefficients of the wavelets series representation of HFSM. Our methodology can be extended to more general harmonizable stable processes and fields.
There are two classical very different extensions of the well-known Gaussian fractional Brownian ... more There are two classical very different extensions of the well-known Gaussian fractional Brownian motion to non-Gaussian frameworks of heavy-tailed stable distributions: the harmonizable fractional stable motion (HFSM) and the linear fractional stable motion (LFSM). As far as we know, while several articles in the literature, some of which appeared a long time ago, have proposed statistical estimators for parameters of LFSM, no estimator has yet been proposed in the framework of HFSM. Among other things, what makes statistical estimation of parameters of HFSM to be a difficult problem is that, in contrast to LFSM, HFSM is not ergodic. The main goal of our work is to propose a new strategy for dealing with this problem and constructing strongly consistent and asymptotically normal statistical estimators for both parameters of HFSM. The keystone of our new strategy consists in the construction of new transforms of HFSM which allow to obtain, at any dyadic level and also at any two consecutive dyadic levels, sequences of independent stable random variables. This new strategy might allow to make statistical inference for more general non-ergodic hamonizable stable processes and fields than HFSM. Moreover, it could turn out to be useful in study of other issues related to them.
There are two classical very different extensions of the well-known Gaussian fractional Brownian ... more There are two classical very different extensions of the well-known Gaussian fractional Brownian motion to non-Gaussian frameworks of heavy-tailed stable distributions: the harmonizable fractional stable motion (HFSM) and the linear fractional stable motion (LFSM). As far as we know, while several articles in the literature, some of which appeared a long time ago, have proposed statistical estimators for the parameters of LFSM, no estimator has yet been proposed in the framework of HFSM. Among other things, what makes statistical estimation of parameters of HFSM to be a difficult problem is that, in contrast to LFSM, HFSM is not ergodic. The main goal of our work is to propose a new strategy for dealing with this problem and obtaining solutions of it. The keystone of our new strategy consists in the construction of new transforms of HFSM which allow to obtain, at any dyadic level, a sequence of independent stable random variables.
Multistable distributions are natural extensions of symmetric α stable distributions. They have b... more Multistable distributions are natural extensions of symmetric α stable distributions. They have been introduced quite recently by Falconer, Lévy Véhel and their co-authors in [1, 2, 3]. Roughly speaking such a distribution is obtained by replacing the constant parameter α of a symmetric stable distribution by a (Lebesgue) mesurable function α(x) with values in [a, 2], where a > 0 denotes a fixed arbitrarily small real number. Let Y be an arbitrary symmetric α stable random variable of scale parameter σ > 0, an important classical result concerning the heavy-tailed behavior of its distribution (see e.g. [4]) is that there exists an explicit constant C(α) > 0, only depending on α ∈ (0, 2), such that lim λ→+∞ C(α)σ α λ −α −1 P |Y | > λ = 1. In this article, we show that the latter result can be extended to the setting of multistable random variables, when the function α(x) is with values in an arbitray compact interval [a, b] contained in (0, 2).
The most known example of a class of non-Gaussian stochastic processes which belongs to the homog... more The most known example of a class of non-Gaussian stochastic processes which belongs to the homogenous Wiener chaos of an arbitrary order N > 1 are probably Hermite processes of rank N. They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound of the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N , which, among other things, shows that these paths are nowhere differentiable functions.
The study of path behaviour of stochastic processes is a classical topic in probability theory an... more The study of path behaviour of stochastic processes is a classical topic in probability theory and related areas. In this frame, a natural question one can address is: whether or not sample paths belong to a critical Hölder space? The answer to this question is negative in the case of Brownian motion and many other stochastic processes: it is well-known that despite the fact that Brownian paths satisfy, on each compact interval I, a Hölder condition of any order strictly less than 1/2, they fail to belong to the critical Hölder space C 1/2 (I). In this article, we show that a different phenomenon happens in the case of linear multifractional stable motion (LMSM): for any given compact interval one can find a critical Hölder space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived in (Biermé, H. and Lacaux, C., 2013), on behaviour of LMSM, by showing that the logarithmic factor in it is not needed.
We consider a modified quadratic variation of the Hermite process based on some well-chosen incre... more We consider a modified quadratic variation of the Hermite process based on some well-chosen increments of this process. These special increments have the very useful property to be independent and identically distributed up to asymptotically negligible remainders. We prove that this modified quadratic variation satisfies a Central Limit Theorem and we derive its rate of convergence under the Wasserstein distance via Stein-Malliavin calculus. As a consequence, we construct, for the first time in the literature related to Hermite processes, a strongly consistent and asymptotically normal estimator for the Hurst parameter.
Spectral singularities at non-zero frequencies play an important role in investigating cyclic or ... more Spectral singularities at non-zero frequencies play an important role in investigating cyclic or seasonal time series. The publication [2] introduced the generalized filtered method-of-moments approach to simultaneously estimate singularity location and long-memory parameters. This paper continues studies of these simultaneous estimators. A wide class of Gegenbauer-type semi-parametric models is considered. Asymptotic normality of several statistics of the cyclic and long-memory parameters is proved. New adjusted estimates are proposed and investigated. The theoretical findings are illustrated by numerical results. The methodology includes wavelet transformations as a particular case.
A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years... more A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in [2] by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. It consists in substituting to the Hurst parameter, in a stochastic integral representation of the high-frequency part of FBM, a random variable depending on the integration variable. The MPRE obtained in this way offers, among other things, the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [2]. Yet, the study of Hölder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis.
HAL (Le Centre pour la Communication Scientifique Directe), May 28, 2022
Multifractional processes are extensions of Fractional Brownian Motion obtained by replacing its ... more Multifractional processes are extensions of Fractional Brownian Motion obtained by replacing its constant Hurst parameter by a deterministic or a random function H(•), called the Hurst function, which allows to prescribe their local sample paths roughness at each point. For that reason statistical estimation of H(•) is an important issue. Many articles have dealt with this issue in the case where * Corresponding author consistency result in uniform norm is rather unusual in literature on statistical estimation of functions.
Linear fractional stable motion, denoted by {X H,α (t)} t∈R , is one of the most classical stable... more Linear fractional stable motion, denoted by {X H,α (t)} t∈R , is one of the most classical stable processes; it depends on two parameters H ∈ (0, 1) and α ∈ (0, 2). The parameter H characterizes the self-similarity property of {X H,α (t)} t∈R while the parameter α governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that H > 1/α and that H is known. We show that, on the interval [0, 1], the asymptotic behaviour of the maximum, at a given scale j, of absolute values of the wavelet coefficients of {X H,α (t)} t∈R , is of the same order as 2 −j(H−1/α) ; then we derive from this result a strongly consistent (i.e. almost surely convergent) statistical estimator for the parameter α.
Springer series in computational neuroscience, 2016
The characteristics of biomedical signals are not captured by conventional measures like the aver... more The characteristics of biomedical signals are not captured by conventional measures like the average amplitude of the signal. The methodologies derived from fractal geometry have been a very useful approach to study the degree of irregularity of a signal. The monofractal analysis of a signal is defined by a single power-law exponent in assuming a scale invariance in time and space. However, temporal and spatial variation in scale invariant structure of the biomedical signal often appears. In this case, the multifractal analysis is well suited because it is defined by a multifractal spectrum of power-law exponents. There are several approaches to the implementation of this analysis and there are numerous ways to present these.
Une base d'ondelettes orthonormales dyadiques de L#2 (R#D) est une base hilbertienne de la fo... more Une base d'ondelettes orthonormales dyadiques de L#2 (R#D) est une base hilbertienne de la forme 2#J#D#/#2#I(2#JX K)/ I , 1,, 2#D 1, J , Z et K , Z#D, elle est en general associee a une analyse multi resolution et a un banc de filtres. On ne savait pas encore construire des ondelettes meres #I non separables, a support compact et de regularite quelconque, (une ondelette separable est un produit d'ondelette(s) et de fonction(s) d'echelle monodimensionnelles). L'objectif de cette these est d'analyser et de resoudre ce probleme. Nous etablissons d'abord, au moyen d'outils de geometrie algebrique, que la plupart des analyses multi resolution dont le QMF est d'une taille donnee sont non separables. Nous construisons ensuite, par des calculs assez simples, de nouveaux bancs de filtres, le plus souvent non separables. Nous montrons enfin que certains de ces bancs de filtres engendrent des bases d'ondelettes de I#2 (R#D), dyadiques, orthonormales, non separables, a support compact et de regularite arbitrairement elevee.
The aim of this paper is to prove that wavelet leaders allow to get very fine properties of the t... more The aim of this paper is to prove that wavelet leaders allow to get very fine properties of the trajectories of the Brownian motion: we show that the three well-known behaviors of its oscillations, namely to be ordinary, rapid and slow, are also present in the behavior of the size of its wavelet leaders.
Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and... more Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and many other related stochastic processes and fields have started to be introduced since more than two decades. Such representations provide natural frameworks for approximating almost surely and uniformly rough sample paths at different scales and for study of various aspects of their complex erratic behavior. Hermite process of an arbitrary integer order d, which extends FBM, is a paradigmatic example of a stochastic process belonging to the dth Wiener chaos. It was introduced very long time ago, yet many of its properties are still unknown when d ≥ 3. In a paper published in 2004, Pipiras raised the problem to know whether wavelet-type random series representations with a well-localized smooth scaling function, reminiscent to those for FBM due to Meyer, Sellan and Taqqu, can be obtained for a Hermite process of any order d. He solved it in this same paper in the particular case d = 2 in which the Hermite process is called the Rosenblatt process. Yet, the problem remains unsolved in the general case d ≥ 3. The main goal of our article is to solve it, not only for usual Hermite processes but also for generalizations of them. Another important goal of our article is to derive almost sure uniform estimates of the errors related with approximations of such processes by scaling functions parts of their wavelet-type random series representations.
The characteristics of biomedical signals are not captured by conventional measures like the aver... more The characteristics of biomedical signals are not captured by conventional measures like the average amplitude of the signal. The methodologies derived from fractal geometry have been a very useful approach to study the degree of irregularity of a signal. The monofractal analysis of a signal is defined by a single power-law exponent in assuming a scale invariance in time and space. However, temporal and spatial variation in scale invariant structure of the biomedical signal often appears. In this case, the multifractal analysis is well suited because it is defined by a multifractal spectrum of power-law exponents. There are several approaches to the implementation of this analysis and there are numerous ways to present these.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 30, 2021
In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Ra... more In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on an universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions.
Multifractional processes are stochastic processes with non-stationary increments whose local reg... more Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} . In our article, under a rather weak condition on the functional para...
In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Ra... more In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on an universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions.
Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of... more Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of the well-known Fractional Brownian Motion to the framework of heavytailed stable distributions. It was introduced several decades ago; however its properties are far from being completely understood. In our present paper we determine the optimal power of the logarithmic factor in a uniform modulus of continuity for HFSM, which solves an open old problem. The keystone of our strategy consists in Abel transforms of the LePage series expansions of the random coefficients of the wavelets series representation of HFSM. Our methodology can be extended to more general harmonizable stable processes and fields.
There are two classical very different extensions of the well-known Gaussian fractional Brownian ... more There are two classical very different extensions of the well-known Gaussian fractional Brownian motion to non-Gaussian frameworks of heavy-tailed stable distributions: the harmonizable fractional stable motion (HFSM) and the linear fractional stable motion (LFSM). As far as we know, while several articles in the literature, some of which appeared a long time ago, have proposed statistical estimators for parameters of LFSM, no estimator has yet been proposed in the framework of HFSM. Among other things, what makes statistical estimation of parameters of HFSM to be a difficult problem is that, in contrast to LFSM, HFSM is not ergodic. The main goal of our work is to propose a new strategy for dealing with this problem and constructing strongly consistent and asymptotically normal statistical estimators for both parameters of HFSM. The keystone of our new strategy consists in the construction of new transforms of HFSM which allow to obtain, at any dyadic level and also at any two consecutive dyadic levels, sequences of independent stable random variables. This new strategy might allow to make statistical inference for more general non-ergodic hamonizable stable processes and fields than HFSM. Moreover, it could turn out to be useful in study of other issues related to them.
There are two classical very different extensions of the well-known Gaussian fractional Brownian ... more There are two classical very different extensions of the well-known Gaussian fractional Brownian motion to non-Gaussian frameworks of heavy-tailed stable distributions: the harmonizable fractional stable motion (HFSM) and the linear fractional stable motion (LFSM). As far as we know, while several articles in the literature, some of which appeared a long time ago, have proposed statistical estimators for the parameters of LFSM, no estimator has yet been proposed in the framework of HFSM. Among other things, what makes statistical estimation of parameters of HFSM to be a difficult problem is that, in contrast to LFSM, HFSM is not ergodic. The main goal of our work is to propose a new strategy for dealing with this problem and obtaining solutions of it. The keystone of our new strategy consists in the construction of new transforms of HFSM which allow to obtain, at any dyadic level, a sequence of independent stable random variables.
Multistable distributions are natural extensions of symmetric α stable distributions. They have b... more Multistable distributions are natural extensions of symmetric α stable distributions. They have been introduced quite recently by Falconer, Lévy Véhel and their co-authors in [1, 2, 3]. Roughly speaking such a distribution is obtained by replacing the constant parameter α of a symmetric stable distribution by a (Lebesgue) mesurable function α(x) with values in [a, 2], where a > 0 denotes a fixed arbitrarily small real number. Let Y be an arbitrary symmetric α stable random variable of scale parameter σ > 0, an important classical result concerning the heavy-tailed behavior of its distribution (see e.g. [4]) is that there exists an explicit constant C(α) > 0, only depending on α ∈ (0, 2), such that lim λ→+∞ C(α)σ α λ −α −1 P |Y | > λ = 1. In this article, we show that the latter result can be extended to the setting of multistable random variables, when the function α(x) is with values in an arbitray compact interval [a, b] contained in (0, 2).
The most known example of a class of non-Gaussian stochastic processes which belongs to the homog... more The most known example of a class of non-Gaussian stochastic processes which belongs to the homogenous Wiener chaos of an arbitrary order N > 1 are probably Hermite processes of rank N. They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound of the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N , which, among other things, shows that these paths are nowhere differentiable functions.
The study of path behaviour of stochastic processes is a classical topic in probability theory an... more The study of path behaviour of stochastic processes is a classical topic in probability theory and related areas. In this frame, a natural question one can address is: whether or not sample paths belong to a critical Hölder space? The answer to this question is negative in the case of Brownian motion and many other stochastic processes: it is well-known that despite the fact that Brownian paths satisfy, on each compact interval I, a Hölder condition of any order strictly less than 1/2, they fail to belong to the critical Hölder space C 1/2 (I). In this article, we show that a different phenomenon happens in the case of linear multifractional stable motion (LMSM): for any given compact interval one can find a critical Hölder space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived in (Biermé, H. and Lacaux, C., 2013), on behaviour of LMSM, by showing that the logarithmic factor in it is not needed.
We consider a modified quadratic variation of the Hermite process based on some well-chosen incre... more We consider a modified quadratic variation of the Hermite process based on some well-chosen increments of this process. These special increments have the very useful property to be independent and identically distributed up to asymptotically negligible remainders. We prove that this modified quadratic variation satisfies a Central Limit Theorem and we derive its rate of convergence under the Wasserstein distance via Stein-Malliavin calculus. As a consequence, we construct, for the first time in the literature related to Hermite processes, a strongly consistent and asymptotically normal estimator for the Hurst parameter.
Spectral singularities at non-zero frequencies play an important role in investigating cyclic or ... more Spectral singularities at non-zero frequencies play an important role in investigating cyclic or seasonal time series. The publication [2] introduced the generalized filtered method-of-moments approach to simultaneously estimate singularity location and long-memory parameters. This paper continues studies of these simultaneous estimators. A wide class of Gegenbauer-type semi-parametric models is considered. Asymptotic normality of several statistics of the cyclic and long-memory parameters is proved. New adjusted estimates are proposed and investigated. The theoretical findings are illustrated by numerical results. The methodology includes wavelet transformations as a particular case.
A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years... more A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in [2] by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. It consists in substituting to the Hurst parameter, in a stochastic integral representation of the high-frequency part of FBM, a random variable depending on the integration variable. The MPRE obtained in this way offers, among other things, the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [2]. Yet, the study of Hölder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis.
HAL (Le Centre pour la Communication Scientifique Directe), May 28, 2022
Multifractional processes are extensions of Fractional Brownian Motion obtained by replacing its ... more Multifractional processes are extensions of Fractional Brownian Motion obtained by replacing its constant Hurst parameter by a deterministic or a random function H(•), called the Hurst function, which allows to prescribe their local sample paths roughness at each point. For that reason statistical estimation of H(•) is an important issue. Many articles have dealt with this issue in the case where * Corresponding author consistency result in uniform norm is rather unusual in literature on statistical estimation of functions.
Linear fractional stable motion, denoted by {X H,α (t)} t∈R , is one of the most classical stable... more Linear fractional stable motion, denoted by {X H,α (t)} t∈R , is one of the most classical stable processes; it depends on two parameters H ∈ (0, 1) and α ∈ (0, 2). The parameter H characterizes the self-similarity property of {X H,α (t)} t∈R while the parameter α governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that H > 1/α and that H is known. We show that, on the interval [0, 1], the asymptotic behaviour of the maximum, at a given scale j, of absolute values of the wavelet coefficients of {X H,α (t)} t∈R , is of the same order as 2 −j(H−1/α) ; then we derive from this result a strongly consistent (i.e. almost surely convergent) statistical estimator for the parameter α.
Springer series in computational neuroscience, 2016
The characteristics of biomedical signals are not captured by conventional measures like the aver... more The characteristics of biomedical signals are not captured by conventional measures like the average amplitude of the signal. The methodologies derived from fractal geometry have been a very useful approach to study the degree of irregularity of a signal. The monofractal analysis of a signal is defined by a single power-law exponent in assuming a scale invariance in time and space. However, temporal and spatial variation in scale invariant structure of the biomedical signal often appears. In this case, the multifractal analysis is well suited because it is defined by a multifractal spectrum of power-law exponents. There are several approaches to the implementation of this analysis and there are numerous ways to present these.
Une base d'ondelettes orthonormales dyadiques de L#2 (R#D) est une base hilbertienne de la fo... more Une base d'ondelettes orthonormales dyadiques de L#2 (R#D) est une base hilbertienne de la forme 2#J#D#/#2#I(2#JX K)/ I , 1,, 2#D 1, J , Z et K , Z#D, elle est en general associee a une analyse multi resolution et a un banc de filtres. On ne savait pas encore construire des ondelettes meres #I non separables, a support compact et de regularite quelconque, (une ondelette separable est un produit d'ondelette(s) et de fonction(s) d'echelle monodimensionnelles). L'objectif de cette these est d'analyser et de resoudre ce probleme. Nous etablissons d'abord, au moyen d'outils de geometrie algebrique, que la plupart des analyses multi resolution dont le QMF est d'une taille donnee sont non separables. Nous construisons ensuite, par des calculs assez simples, de nouveaux bancs de filtres, le plus souvent non separables. Nous montrons enfin que certains de ces bancs de filtres engendrent des bases d'ondelettes de I#2 (R#D), dyadiques, orthonormales, non separables, a support compact et de regularite arbitrairement elevee.
The aim of this paper is to prove that wavelet leaders allow to get very fine properties of the t... more The aim of this paper is to prove that wavelet leaders allow to get very fine properties of the trajectories of the Brownian motion: we show that the three well-known behaviors of its oscillations, namely to be ordinary, rapid and slow, are also present in the behavior of the size of its wavelet leaders.
Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and... more Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and many other related stochastic processes and fields have started to be introduced since more than two decades. Such representations provide natural frameworks for approximating almost surely and uniformly rough sample paths at different scales and for study of various aspects of their complex erratic behavior. Hermite process of an arbitrary integer order d, which extends FBM, is a paradigmatic example of a stochastic process belonging to the dth Wiener chaos. It was introduced very long time ago, yet many of its properties are still unknown when d ≥ 3. In a paper published in 2004, Pipiras raised the problem to know whether wavelet-type random series representations with a well-localized smooth scaling function, reminiscent to those for FBM due to Meyer, Sellan and Taqqu, can be obtained for a Hermite process of any order d. He solved it in this same paper in the particular case d = 2 in which the Hermite process is called the Rosenblatt process. Yet, the problem remains unsolved in the general case d ≥ 3. The main goal of our article is to solve it, not only for usual Hermite processes but also for generalizations of them. Another important goal of our article is to derive almost sure uniform estimates of the errors related with approximations of such processes by scaling functions parts of their wavelet-type random series representations.
The characteristics of biomedical signals are not captured by conventional measures like the aver... more The characteristics of biomedical signals are not captured by conventional measures like the average amplitude of the signal. The methodologies derived from fractal geometry have been a very useful approach to study the degree of irregularity of a signal. The monofractal analysis of a signal is defined by a single power-law exponent in assuming a scale invariance in time and space. However, temporal and spatial variation in scale invariant structure of the biomedical signal often appears. In this case, the multifractal analysis is well suited because it is defined by a multifractal spectrum of power-law exponents. There are several approaches to the implementation of this analysis and there are numerous ways to present these.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 30, 2021
In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Ra... more In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on an universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions.
Multifractional processes are stochastic processes with non-stationary increments whose local reg... more Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} . In our article, under a rather weak condition on the functional para...
In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Ra... more In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on an universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions.
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Papers by Antoine Ayache