arXiv subjects

Paul Mansanarez

Publications and source records attributed to Paul Mansanarez.

5 recordsLinked to original sources

arXiv2026

Breuer-Major-Donsker invariance principle

We prove a Breuer-Major-type Donsker's invariance principle for stationary Gaussian sequences under the natural \emph{finite-variance assumption} on the test function. This result, which we call the \emph{Breuer-Major-Donsker} principle, or simply the \emph{BMD principle}, removes the additional moment assumption imposed in the functional Breuer-Major theorem of Nourdin and Nualart (\emph{Probab. Theory Related Fields}, 2020). Our method does not rely on the Malliavin-calculus estimates used by Nourdin and Nualart, in particular Meyer's inequality. Instead, it is based on a predictable-martingale decomposition of the partial-sum process, which is of independent interest. We also make systematic use of \emph{non-determinism}, a central notion in Gaussian prediction theory. In the non-deterministic case, the martingale part is handled by the martingale functional central limit theorem, while the predictable remainder gains integrability above order two through Ornstein-Uhlenbeck smoothing. In the deterministic case, the martingale part vanishes, and the smoothing mechanism is no longer available along the full sequence. Nevertheless, under an additional mild assumption on the covariance function, a suitable decimation recovers non-determinism and reduces the proof to the non-deterministic case.

math.PR↗

arXiv2025

Edgeworth expansion on Wiener chaos

Consider $F$ an element of the $p$-th Wiener chaos $\WW_p$, and denote by $\prob_F$ its law. For a positive integer $m$, let $\boldsymbol{\gamma}_{F,m}$ be the Radon measure with density $x \mapsto \frac{e^{-x^2/2}}{\sqrt{2\pi}} \left(1 + \sum_{k=3}^{4m-1} \frac{\E[H_k(F)]}{k!}\, H_k(x)\right)$, where $H_k$ is the $k$-th Hermite polynomial. The main goal of this article is to prove that the total variation distance between $\prob_F$ and $\boldsymbol{\gamma}_{F,m}$ is of order $\Var(\Gamma(F,F))^{({m+1})/{2}}$, where $\Gamma(F,F)$ denotes the carr\'e-du-champ operator of $F$. The variance of $\Gamma(F,F)$ is known to govern Gaussian fluctuations and can be bounded from above by $\kappa_4(F)$, the fourth cumulant of $F$, as established in the seminal work \cite{NP2009a}. Our result thus provides a genuine Edgeworth expansion in the setting of central convergence on Wiener chaoses. In this context, the quantity $\Var(\Gamma(F,F))$ plays the role of the small parameter that governs the accuracy of the approximation, in the same way that $1/\sqrt{n}$ does in the classical central limit theorem. To the best of our knowledge, our work is the first to establish Edgeworth expansions for Wiener chaoses in full generality and at arbitrary order, together with explicit remainder bounds that systematically improve with the order of the expansion--exactly as one would expect from an Edgeworth approximation. Our results apply verbatim to every situation where a central limit theorem is available for chaos elements, since no structural assumption is required beyond belonging to a fixed Wiener chaos. As a byproduct, we recover the celebrated optimal fourth moment theorem from \cite{NP2015} by combining the expansions at the first and second orders, with sharper quantitative bounds. Previous works on Edgeworth expansions for Wiener chaoses were essentially restricted to the first order.

math.PR↗

arXiv2025

Stein's method for Fr\'echet approximation: a regularly varying functions approach

We develop a variant of Stein's method of comparison of generators to bound the Kolmogorov, total variation, and Wasserstein-1 distances between distributions on the real line. Our discrepancy is expressed in terms of the ratio of reverse hazard rates; it therefore remains tractable even when density derivatives are intractable. Our main application concerns the approximation of normalized extremes by Fr\'echet laws. In this setting, the new discrepancy provides a quantitative measure of distributional proximity in terms of the average regular variation at infinity of the underlying cumulative distribution function. We illustrate the approach through explicit computations for maxima of Pareto, Cauchy, and Burr~XII distributions. Our new discrepancy also opens the way to statistical applications which we outline.

math.PR↗

arXiv2025

Non-separable graphs meet Ledoux's polynomials

In the pathbreaking article \cite{LED16}, an integral representation of the derivatives of entropy along the heat flow of a probability measure was established under suitable moment conditions. These integral representations have found significant applications in diverse domains - notably in information theory (e.g., entropy power inequalities, monotonicity of Fisher information) and in estimation theory (through the link between entropy derivatives and the minimum mean square error, MMSE, in Gaussian channels). The representations involve multivariate polynomials $(R_n)_n$, arising from a Lie algebra framework on multilinear operators. Despite their central role, the combinatorial structure of these polynomials remains only partially understood. In this note, we prove that the number of monomials in $R_n$ coincides with the number of degree sequences with degree sum $2n$ having a non-separable graph realization, thereby resolving a conjecture from \cite{MPS24}, and drawing an interesting link between these two domains.

math.CO↗

arXiv2024

Derivatives of entropy and the MMSE conjecture

We investigate the entropy $H(μ,t)$ of a probability measure $μ$ along the heat flow and more precisely we seek for closed algebraic representations of its derivatives. Provided that $μ$ admits moments of any order, it is indeed proved in [Guo et al., 2010] that $t\mapsto H(μ,t)$ is smooth, and in [Ledoux, 2016] that its derivatives at zero can be expressed into multivariate polynomials evaluated in the moments (or cumulants) of $μ$. In the seminal contribution \cite{Led}, these algebraic expressions are derived through $Γ$-calculus techniques which provide implicit recursive formulas for these polynomials. Our main contribution consists in a fine combinatorial analysis of these inductive relations and for the first time to derive closed formulas for the leading coefficients of these polynomials expressions. Building upon these explicit formulas we revisit the so-called "MMSE conjecture" from [Guo et al., 2010] which asserts that two distributions on the real line with the same entropy along the heat flow must coincide up to translation and symmetry. Our approach enables us to provide new conditions on the source distributions ensuring that the MMSE conjecture holds and to refine several criteria proved in [Ledoux, 2016]. As illustrating examples, our findings cover the cases of uniform and Rademacher distributions, for which previous results in the literature were inapplicable.

cs.IT↗