arXiv subjects

Marie-Christine Düker

Publications and source records attributed to Marie-Christine Düker.

16 recordsLinked to original sources

arXiv2026

Breuer-Major Theorems for Hilbert Space-Valued Random Variables

Let $\{X_k\}_{k\in\mathbb Z}$ be a stationary Gaussian process with values in a separable Hilbert space $\mathcal H_1$, and let $G:\mathcal H_1\to\mathcal H_2$ be a measurable map into another separable Hilbert space $\mathcal H_2$. We derive a central limit theorem for the centered normalized partial sums of the Hilbert space-valued subordinated process $\{G[X_k]\}_{k\in\mathbb Z}$. Our result holds under either of two sets of sufficient conditions, formulated in terms of the transformation $G$ and the temporal and cross-sectional dependence structure of $\{X_k\}_{k\in\mathbb Z}$. These conditions coincide in finite dimensions but lead to genuinely different phenomena in the infinite-dimensional setting. The proof relies on the recently developed Fourth Moment Theorem on Hilbert spaces, leveraging tools from the infinite-dimensional Malliavin-Stein framework. We also provide continuous-time and quantitative versions of the central limit theorem. In a series of examples, we recover and strengthen limit theorems for a wide array of statistics relevant in functional data analysis, and present, as an application of our result, a novel limit theorem in the framework of neural operators.

math.PR↗

arXiv2026

The Fourth-Moment Theorem on Hilbert Spaces

In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-Itô chaos representation to a Gaussian measure on a separable Hilbert space. Our first main result shows that, assuming convergence of the associated covariance operators in the trace-class norm, a sequence lying in a fixed Wiener-Itô chaos converges in distribution if and only if its fourth weak moments converge to the corresponding Gaussian moments. For general sequences with infinite chaos expansions, we derive analogous sufficient conditions for convergence in distribution. A key ingredient in our approach is a Stein-Malliavin bound formulated with respect to a distance that metrizes weak convergence of probability measures on separable Hilbert spaces. The results are infinite-dimensional extensions of the classical real-valued Fourth-Moment Theorem of Nualart and Peccati [Ann. Probab. 33, 177-193 (2005)]. Our work builds upon the work by Bourguin and Campese [Electron. J. Probab. 25, 1-30 (2020)] who claimed a Fourth-Moment Theorem in separable Hilbert spaces. However, a recent work by Bassetti, Bourguin, Campese, and Peccati [Stat. Probab. Lett. 233, 110671, (2026)] showed that the distance employed in the former article does not metrize weak convergence of probability measures on separable Hilbert spaces. Consequently, the conditions stated in Bourguin and Campese are not sufficient to recover a valid Fourth-Moment Theorem in the Hilbert-space setting.

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arXiv2026

RKHS-Based Inference for Nonlinear Granger Causality via Conditional Centering

Granger causality is commonly formulated through linear prediction in vector autoregressive models, limiting its ability to detect nonlinear predictive relationships. We propose a reproducing kernel Hilbert space (RKHS)-based test for nonlinear Granger non-causality in conditional mean for nonlinear autoregressive processes. The key idea is a conditional-centering decomposition of the target regression function into an own-history component and an orthogonal component capturing the additional predictive contribution of the potential source history. Non-causality is characterized by the vanishing of the latter component. The own-history component is estimated by kernel ridge regression, and the residuals are embedded in a second RKHS using a conditionally centered kernel. This yields an RKHS-valued residual moment whose squared norm forms the test statistic. We establish a weighted chi-square null limit and consistency against fixed alternatives for the population-centered statistic. An empirically centered version is shown to retain the null limit and enables spectral calibration of critical values and $p$-values without resampling. Simulation studies demonstrate accurate size control and power against nonlinear alternatives, and real-data applications illustrate the usefulness of the method for detecting nonlinear predictive relationships in time series.

stat.ME↗

arXiv2026

Analytic Standard Errors for Latent Gaussian Discrete-Valued Multivariate Time Series

Unlike their continuous-valued counterparts, there are no universally preferred methodologies for modeling discrete-valued time series. This is especially problematic in fields such as psychology and education, where repeated-measures data often take the form of count, dichotomous, and ordered categorical variables. To address the need for flexible methodology for analyzing discrete-valued time series data, a copula-style multivariate model defined through deterministic functions of a latent stationary Gaussian vector series has been proposed. This model has several promising features, including the ability to accommodate a wide variety of marginal distributions within the same model while also allowing for the most flexible autocorrelation structure possible. We extend this framework by deriving analytic standard errors to facilitate inference on the latent Gaussian dynamics. In so doing, we establish the joint asymptotic normality of estimators of the parameters governing the latent Gaussian series and the marginal distributions. The performance of these analytic standard errors is examined in a simulation study and an empirical application.

stat.ME↗

arXiv2026

Testing common structure in high-dimensional factor models: change-point and two-sample procedures

This work proposes a novel procedure to test for common structures across two high-dimensional factor models. The introduced test allows to uncover whether two factor models are driven by the same loading matrix up to some linear transformation. The test can be used to discover inter-individual relationships between two datasets. In addition, it can be applied to test for structural changes over time in the loading matrix of an individual factor model. The test aims to reduce the set of possible alternatives in a classical change-point setting. The theoretical results establish the asymptotic behavior of the introduced test statistic. The theory is supported by a simulation study showing promising results in empirical test size and power. Two real data applications are considered: the first investigates changes in the loadings of the celebrated US macroeconomic dataset of Stock and Watson, and the second examines similarities of the loadings of macroeconomic indicators for the US and South Korea.

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arXiv2026

Prior distributions for structured semi-orthogonal matrices

Statistical models for multivariate data often include a semi-orthogonal matrix parameter. In many applications, there is reason to expect that the semi-orthogonal matrix parameter satisfies a structural assumption such as sparsity or smoothness. From a Bayesian perspective, these structural assumptions should be incorporated into an analysis through the prior distribution. In this work, we introduce a general approach to constructing prior distributions for structured semi-orthogonal matrices that leads to tractable posterior inference via parameter-expanded Markov chain Monte Carlo. We draw on recent results from random matrix theory to establish a theoretical basis for the proposed approach. We then introduce specific prior distributions for incorporating sparsity or smoothness and illustrate their use through applications to biological and oceanographic data.

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arXiv2025

Testing Simultaneous Diagonalizability

This paper proposes novel methods to test for simultaneous diagonalization of possibly asymmetric matrices. Motivated by various applications, a two-sample test as well as a generalization for multiple matrices are proposed. A partial version of the test is also studied to check whether a partial set of eigenvectors is shared across samples. Additionally, a novel algorithm for the considered testing methods is introduced. Simulation studies demonstrate favorable performance for all designs. Finally, the theoretical results are utilized to decouple vector autoregression models into multiple univariate time series, and to test for the same stationary distribution in recurrent Markov chains. These applications are demonstrated using macroeconomic indices of 8 countries and streamflow data, respectively.

stat.ME↗

arXiv2025

Scaling limits for sample autocovariance operators of Hilbert space-valued linear processes

This article considers linear processes with values in a separable Hilbert space exhibiting long-range dependence. The scaling limits for the sample autocovariance operators at different time lags are investigated in the topology of their respective Hilbert spaces. Distinguishing two different regimes of long-range dependence, the limiting object is either a Hilbert space-valued Gaussian or a Hilbert space-valued non-Gaussian random variable. The latter can be represented as a unitary transformation of double Wiener-Itô integrals with sample paths in a function space. This work is the first to show weak convergence to such double stochastic integrals with sample paths in infinite dimensions. The result generalizes the well known convergence to a Hermite process in finite dimensions, introducing a new domain of attraction for probability measures in Hilbert spaces. The key technical contributions include the introduction of double Wiener-Itô integrals with values in a function space and with dependent integrators, as well as establishing sufficient conditions for their existence as limits of sample autocovariance operators.

math.PR↗

arXiv2025

Higher-order approximation for uncertainty quantification in time series analysis

For time series with high temporal correlation, the empirical process converges rather slowly to its limiting distribution. Many statistics in change-point analysis, goodness-of-fit testing and uncertainty quantification admit a representation as functionals of the empirical process and therefore inherit its slow convergence. As a result, inference based on the asymptotic distribution of those quantities is significantly affected by relatively small sample sizes. We assess the quality of higher-order approximations of the empirical process by deriving the asymptotic distribution of the corresponding error terms. Based on the limiting distribution of the higher-order terms, we propose a novel approach to calculate confidence intervals for statistical quantities such as the median. In a simulation study, we compare coverage rates and lengths of these confidence intervals with those based on the asymptotic distribution of the empirical process and highlight some benefits of higher-order approximations of the empirical process.

math.ST↗

arXiv2025

Latent Gaussian dynamic factor modeling and forecasting for multivariate count time series

This work considers estimation and forecasting in a multivariate, possibly high-dimensional count time series model constructed from a transformation of a latent Gaussian dynamic factor series. The estimation of the latent model parameters is based on second-order properties of the count and underlying Gaussian time series, yielding estimators of the underlying covariance matrices for which standard principal component analysis applies. Theoretical consistency results are established for the proposed estimation, building on certain concentration results for the models of the type considered. They also involve the memory of the latent Gaussian process, quantified through a spectral gap, shown to be suitably bounded as the model dimension increases, which is of independent interest. In addition, novel cross-validation schemes are suggested for model selection. The forecasting is carried out through a particle-based sequential Monte Carlo, leveraging Kalman filtering techniques. A simulation study and an application are also considered.

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arXiv2025

Kernel Estimation for Nonlinear Dynamics

Many scientific problems involve data exhibiting both temporal and cross-sectional dependencies. While linear dependencies have been extensively studied, the theoretical analysis of regression estimators under nonlinear dependencies remains scarce. This work studies a kernel-based estimation procedure for nonlinear dynamics within the reproducing kernel Hilbert space framework, focusing on nonlinear vector autoregressive models. We derive nonasymptotic probabilistic bounds on the deviation between a regularized kernel estimator and the nonlinear regression function. A key technical contribution is a concentration bound for quadratic forms of stochastic matrices in the presence of dependent data, which is of independent interest. Additionally, we characterize conditions on multivariate kernels that guarantee optimal convergence rates.

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arXiv2024

Vector AutoRegressive Moving Average Models: A Review

Vector AutoRegressive Moving Average (VARMA) models form a powerful and general model class for analyzing dynamics among multiple time series. While VARMA models encompass the Vector AutoRegressive (VAR) models, their popularity in empirical applications is dominated by the latter. Can this phenomenon be explained fully by the simplicity of VAR models? Perhaps many users of VAR models have not fully appreciated what VARMA models can provide. The goal of this review is to provide a comprehensive resource for researchers and practitioners seeking insights into the advantages and capabilities of VARMA models. We start by reviewing the identification challenges inherent to VARMA models thereby encompassing classical and modern identification schemes and we continue along the same lines regarding estimation, specification and diagnosis of VARMA models. We then highlight the practical utility of VARMA models in terms of Granger Causality analysis, forecasting and structural analysis as well as recent advances and extensions of VARMA models to further facilitate their adoption in practice. Finally, we discuss some interesting future research directions where VARMA models can fulfill their potentials in applications as compared to their subclass of VAR models.

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arXiv2023

High-dimensional latent Gaussian count time series: Concentration results for autocovariances and applications

This work considers stationary vector count time series models defined via deterministic functions of a latent stationary vector Gaussian series. The construction is very general and ensures a pre-specified marginal distribution for the counts in each dimension, depending on unknown parameters that can be marginally estimated. The vector Gaussian series injects flexibility into the model's temporal and cross-dimensional dependencies, perhaps through a parametric model akin to a vector autoregression. We show that the latent Gaussian model can be estimated by relating the covariances of the counts and the latent Gaussian series. In a possibly high-dimensional setting, concentration bounds are established for the differences between the estimated and true latent Gaussian autocovariances, in terms of those for the observed count series and the estimated marginal parameters. The results are applied to the case where the latent Gaussian series is a vector autoregression, and its parameters are estimated sparsely through a LASSO-type procedure.

math.ST↗

arXiv2022

Local Whittle estimation of high-dimensional long-run variance and precision matrices

This work develops non-asymptotic theory for estimation of the long-run variance matrix and its inverse, the so-called precision matrix, for high-dimensional time series under general assumptions on the dependence structure including long-range dependence. The estimation involves shrinkage techniques which are thresholding and penalizing versions of the classical multivariate local Whittle estimator. The results ensure consistent estimation in a double asymptotic regime where the number of component time series is allowed to grow with the sample size as long as the true model parameters are sparse. The key technical result is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the true model parameters. In particular, it handles simultaneously the estimation of the memory parameters which enter the underlying model. Novel algorithms for the considered procedures are proposed, and a simulation study and a data application are also provided.

math.ST↗

arXiv2020

Limit theorems for multivariate long-range dependent processes

This article considers multivariate linear processes whose components are either short- or long-range dependent. The functional central limit theorems for the sample mean and the sample autocovariances for these processes are investigated, paying special attention to the mixed cases of short- and long-range dependent series. The resulting limit processes can involve multivariate Brownian motion marginals, operator fractional Brownian motions and matrix-valued versions of the so-called Rosenblatt process.

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arXiv2017

Limit theorems for Hilbert space-valued linear processes under long range dependence

Let $(X_{k})_{k \in \mathbb Z }$ be a linear process with values in a separable Hilbert space $\mathbb{H}$ given by $X_{k} =\sum_{j=0}^{\infty} (j+1)^{-N}\varepsilon_{k-j}$ for each $k \in \mathbb Z$, where $N:\mathbb{H} \to \mathbb{H}$ is a bounded, linear normal operator and $(\varepsilon_{k})_{ k \in \mathbb Z }$ is a sequence of independent, identically distributed $\mathbb{H}$-valued random variables with $E\varepsilon_{0}=0$ and $E\| \varepsilon_{0} \|^2<\infty$. We investigate the central and the functional central limit theorem for $(X_{k})_{k \in \mathbb Z }$ when the series of operator norms $\sum_{j=0}^{\infty} \|(j+1)^{-N}\|_{op}$ diverges. Furthermore we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.

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