arXiv subjects
Publications and source records attributed to Benjamin Arras.
We develop a general distributional theory of fractional (an)isotropic Sobolev spaces associated with the non-degenerate symmetric $\alpha$-stable, $\alpha \in (1,2)$, probability measures on $\mathbb{R}^d$.
In these notes, we obtain new stability estimates for centered non-degenerate selfdecomposable probability measures on $\mathbb{R}^d$ with finite second moment and for non-degenerate symmetric $\alpha$-stable probability measures on $\mathbb{R}^d$ with $\alpha \in [1,2)$. These new results are refinements of the corresponding ones available in the literature. The proofs are based on Stein's method for self-decomposable laws, recently developed in a series of papers, and on closed forms techniques together with a new ingredient: weighted Poincar\'e-type inequalities. As applications, rates of convergence in Wasserstein-type distances are computed for several instances of the generalized central limit theorems (CLTs). In particular, a $n^{1-2/\alpha}$-rate is obtained in $1$-Wasserstein distance when the target law is a non-degenerate symmetric $\alpha$-stable one with $\alpha \in (1,2)$. Finally, the non-degenerate symmetric Cauchy case is studied at length from a spectral point of view. At last, in this Cauchy situation, a $n^{-1}$-rate of convergence is obtained when the initial law is a certain instance of layered stable distributions.
We explore connections between covariance representations, Bismut-type formulas and Stein's method. First, using the theory of closed symmetric forms, we derive covariance representations for several well-known probability measures on $\mathbb{R}^d$, $d \geq 1$. When strong gradient bounds are available, these covariance representations immediately lead to $L^p$-$L^q$ covariance estimates, for all $p \in (1, +\infty)$ and $q = p/(p-1)$. Then, we revisit the well-known $L^p$-Poincar\'e inequalities ($p \geq 2$) for the standard Gaussian probability measure on $\mathbb{R}^d$ based on a covariance representation. Moreover, for the nondegenerate symmetric $\alpha$-stable case, $\alpha \in (1,2)$, we obtain $L^p$-Poincar\'e and pseudo-Poincar\'e inequalities, for $p \in (1, \alpha)$, via a detailed analysis of the various Bismut-type formulas at our disposal. Finally, using the construction of Stein's kernels by closed forms techniques, we obtain quantitative high-dimensional CLTs in $1$-Wasserstein distance when the limiting Gaussian probability measure is anisotropic. The dependence on the parameters is completely explicit and the rates of convergence are sharp.
Boundedness properties of operators associated with non-degenerate symmetric $\alpha$-stable, $\alpha \in (1,2)$, probability measures on $\mathbb{R}^d$ are investigated on appropriate, Euclidean or otherwise, $L^p$-spaces, $p \in (1,+\infty)$. Our approach is based on first obtaining Bismut-type formulae which lead to useful representations for various operators. In the Euclidean setting, the method of transference and one-dimensional multiplier theory combined with fine properties of stable distributions provide dimension-free estimates for the fractional Laplacian. In the non-Euclidean setting, we obtain boundedness results for the non-singular cases as well as dimension-free estimates when the reference measure is the rotationally invariant $\alpha$-stable probability measure.
This work explores and develops elements of Stein's method of approximation, in the infinitely divisible setting, and its connections to functional analysis. It is mainly concerned with multivariate self-decomposable laws without finite first moment and, in particular, with $\alpha$-stable ones, $\alpha \in (0,1]$. At first, several characterizations of these laws via covariance identities are presented. In turn, these characterizations lead to integro-differential equations which are solved with the help of both semigroup and Fourier methodologies. Then, Poincar\'e-type inequalities for self-decomposable laws having finite first moment are revisited. In this non-local setting, several algebraic quantities (such as the carr\'e du champs and its iterates) originating in the theory of Markov diffusion operators are computed. Finally, rigidity and stability results for the Poincar\'e-ratio functional of the rotationally invariant $\alpha$-stable laws, $\alpha\in (1,2)$, are obtained; and as such they recover the classical Gaussian setting as $\alpha \to 2$.
We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R}^d$ having finite first moment. Building on previous univariate findings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy specifically designed for infinitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\mathbb{R}^d$ and a non-degenerate self-decomposable target law with finite second moment. Finally, under an appropriate spectral gap assumption, we investigate, via variational methods, the existence of Stein kernels. In particular, this leads to quantitative versions of classical results on characterizations of probability distributions by variational functionals.
We consider the problem of approximating an unknown function $u\in L^2(D,\rho)$ from its evaluations at given sampling points $x^1,\dots,x^n\in D$, where $D\subset \mathbb{R}^d$ is a general domain and $\rho$ is a probability measure. The approximation is picked in a linear space $V_m$ where $m=\dim(V_m)$ and computed by a weighted least squares method. Recent results show the advantages of picking the sampling points at random according to a well-chosen probability measure $\mu$ that depends both on $V_m$ and $\rho$. With such a random design, the weighted least squares approximation is proved to be stable with high probability, and having precision comparable to that of the exact $L^2(D,\rho)$-orthonormal projection onto $V_m$, in a near-linear sampling regime $n\sim{m\log m}$. The present paper is motivated by the adaptive approximation context, in which one typically generates a nested sequence of spaces $(V_m)_{m\geq1}$ with increasing dimension. Although the measure $\mu=\mu_m$ changes with $V_m$, it is possible to recycle the previously generated samples by interpreting $\mu_m$ as a mixture between $\mu_{m-1}$ and an update measure $\sigma_m$. Based on this observation, we discuss sequential sampling algorithms that maintain the stability and approximation properties uniformly over all spaces $V_m$. Our main result is that the total number of computed sample at step $m$ remains of the order $m\log{m}$ with high probability. Numerical experiments confirm this analysis.
We present, in a unified way, a Stein methodology for infinitely divisible laws (without Gaussian component) having finite first moment. Based on a correlation representation, we obtain a characterizing non-local Stein operator which boils down to classical Stein operators in specific examples. Thanks to this characterizing operator, we introduce various extensions of size bias and zero bias distributions and prove that these notions are closely linked to infinite divisibility. Combined with standard Fourier techniques, these extensions also allow obtaining explicit rates of convergence for compound Poisson approximation in particular towards the symmetric $\alpha$-stable distribution. Finally, in the setting of non-degenerate self-decomposable laws, by semigroup techniques, we solve the Stein equation induced by the characterizing non-local Stein operator and obtain quantitative bounds in weak limit theorems for sums of independent random variables going back to the work of Khintchine and L\'evy.
In this paper we propose a new, simple and explicit mechanism allowing to derive Stein operators for random variables whose characteristic function satisfies a simple ODE. We apply this to study random variables which can be represented as linear combinations of (non necessarily independent) gamma distributed random variables. The connection with Malliavin calculus for random variables in the second Wiener chaos is detailed. An application to McKay Type I random variables is also outlined.
We provide a bound on a natural distance between finitely and infinitely supported elements of the unit sphere of $\ell^2(\mathbb{N}^*)$, the space of real valued sequences with finite $\ell^2$ norm. We use this bound to estimate the 2-Wasserstein distance between random variables which can be represented as linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure which is related to Nourdin and Peccati's Malliavin-Stein method. The main area of application of our results is towards the computation of quantitative rates of convergence towards elements of the second Wiener chaos. After particularizing our bounds to this setting and comparing them with the available literature on the subject (particularly the Malliavin-Stein method for Variance-gamma random variables), we illustrate their versatility by tackling three examples: chi-squared approximation for second order $U$-statistics, asymptotics for sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.
In this paper, we combine Hida distribution theory and Sobolev-Watanabe-Kree spaces in order to study finely the link between forward integrals obtained by regularization and Wick-It\^o integrals with respect to fractional Brownian motion and the Rosenblatt process. The new methodology developed in this paper allows to retrieve results for fractional Brownian motion and to obtain new results regarding the Rosenblatt process. In particular, an It\^o formula for functionals of the Rosenblatt process is obtained which holds in the space of square-integrable random variables.
In this paper, we introduce new Stein identities for gamma target distribution as well as a new non-linear channel specifically designed for gamma inputs. From these two ingredients, we derive an explicit and simple formula for the derivative of the input-output mutual information of this non-linear channel with respect to the channel quality parameter. This relation is reminiscent of the well-known link between the derivative of the input-output mutual information of additive Gaussian noise channel with respect to the signal-to-noise ratio and the minimum mean-square error. The proof relies on a rescaled version of De Bruijn identity for gamma target distribution together with a stochastic representation for the gamma specific Fisher information. Finally, we are able to derive precise bounds and asymptotics for the input-output mutual information of the non-linear channel with gamma inputs.
In this paper, we propose a general means of estimating the rate at which convergences in law occur. Our approach, which is an extension of the classical Stein-Tikhomirov method, rests on a new pair of linear operators acting on characteristic functions. In principle, this method is admissible for any approximating sequence and any target, although obviously the conjunction of several favorable factors is necessary in order for the resulting bounds to be of interest. As we briefly discuss, our approach is particularly promising whenever some version of Stein's method applies. We apply our approach to two examples. The first application concerns convergence in law towards targets $F_\infty$ which belong to the second Wiener chaos (i.e. $F_{\infty}$ is a linear combination of independent centered chi-squared rvs). We detail an application to $U$-statistics. The second application concerns convergence towards targets belonging to the generalized Dickman family of distributions. We detail an application to a theorem from number theory. In both cases our method produces bounds of the correct order (up to a logarithmic loss) in terms of quantities which occur naturally in Stein's method.
In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called Stein equations at the heart of Stein's method. We provide several applications, and discuss comparison with recent similar results on the same topic.
We provide the first in-depth study of the "smart path" interpolation between an arbitrary probability measure and the gamma-$(\alpha, \lambda)$ distribution. We propose new explicit representation formulae for the ensuing process as well as a new notion of relative Fisher information with a gamma target distribution. We use these results to prove a differential and an integrated De Bruijn identity which hold under minimal conditions, hereby extending the classical formulae which follow from Bakry, Emery and Ledoux's $\Gamma$-calculus. Exploiting a specific representation of the "smart path", we obtain a new proof of the logarithmic Sobolev inequality for the gamma law with $\alpha\geq 1/2$ as well as a new type of HSI inequality linking relative entropy, Stein discrepancy and standardized Fisher information for the gamma law with $\alpha\geq 1/2$.
In this paper, we define a stochastic calculus with respect to the Rosenblatt process by means of white noise distribution theory. For this purpose, we compute the translated characteristic function of the Rosenblatt process at time $t>0$ in any direction $\xi\in S(\mathbb{R})$ and the derivative of the Rosenblatt process in the white noise sense. Using Wick multiplication by the former derivative and Pettis integration, we define our stochastic integral with respect to the Rosenblatt process for a wide class of distribution processes. We obtain an explicit formula for the variance of such a stochastic integral and It\^o's formulae for a certain class of functionals of the Rosenblatt process. Finally, we compare our stochastic integral to other approaches.
We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local H\"older regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener integrals.
Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian random fields. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball $B(t_0,ρ)$ are bounded from above using the local Hölder exponent at $t_0$. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of the sample path on an open interval are controlled almost surely by the minimum of the local exponents. Then, we apply these generic results to the cases of the multiparameter fractional Brownian motion, the multifractional Brownian motion whose regularity function $H$ is irregular and the generalized Weierstrass function, whose Hausdorff dimensions were unknown so far.