arXiv subjects

Michael Hediger

Publications and source records attributed to Michael Hediger.

4 recordsLinked to original sources

arXiv2025

Equivalent Gaussian distributions on commutative spaces: An RKHS analysis

The investigation of equivalent Gaussian distributions for stochastic processes is a central problem in probability and statistics. In this context, the choice of the index set and the correlation structure, particularly their interaction, plays a crucial role. The purpose of this paper is to show how an explicit description of the corresponding reproducing kernel Hilbert space (RKHS) helps to better understand this interplay. In the stationary setting, when the index set is taken to be a homogeneous space, we show how an RKHS approach allows us to bridge the gap to harmonic analysis on commutative spaces, thereby further complementing the characterization of equivalent Gaussian distributions via their spectral measures.

math.PR↗

arXiv2023

A note on maximal conditional entropy on Lebesgue spaces

Let $(X,\mathcal{B},P)$ be a probability space and $\mathit{a}$ be a sub $\sigma$-field that is generated by an increasing sequence of sub $\sigma$-fields $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Given $\theta \in \Theta$, where $\Theta$ is some set, let $(X_{n}^{\theta})_{n \in \mathbb{N}}$ be a martingale adapted to $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Martin (1969) provides sufficient conditions to show that $(X_{n}^{\theta})_{n \in \mathbb{N}}$ converges a.s. uniformly on $\Theta$ to a random variable $X^{\theta}$. His results are based on the assumption that there exists an integer $n$ s.t. the conditional entropy given $\mathit{a}_{n}$ is uniformly bounded over the set of finite partitions of $X$ with atoms from $\mathit{a}$. This study complements Martin's results by studying the latter assumption on the maximal conditional entropy in the context of measurable partitions of Lebesgue spaces. We provide conditions under which $\mathit{a}$ conveys too much information for the maximal conditional entropy to be finite. As an example, we consider the space of continuous functions with a compact support, equipped with the Borel $\sigma$-field.

math.PR↗

arXiv2022

On the orthogonality of zero-mean Gaussian measures: Sufficiently dense sampling

For a stationary random function $\xi$, sampled on a subset $D$ of $\mathbb{R}^{d}$, we examine the equivalence and orthogonality of two zero-mean Gaussian measures $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ associated with $\xi$. We give the isotropic analog to the result that the equivalence of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ is linked with the existence of a square-integrable extension of the difference between the covariance functions of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ from $D$ to $\mathbb{R}^{d}$. We show that the orthogonality of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ can be recovered when the set of distances from points of $D$ to the origin is dense in the set of non-negative real numbers.

math.PR↗

arXiv2021

Asymptotic analysis of ML-covariance parameter estimators based on covariance approximations

Given a zero-mean Gaussian random field with a covariance function that belongs to a parametric family of covariance functions, we introduce a new notion of likelihood approximations, termed truncated-likelihood functions. Truncated-likelihood functions are based on direct functional approximations of the presumed family of covariance functions. For compactly supported covariance functions, within an increasing-domain asymptotic framework, we provide sufficient conditions under which consistency and asymptotic normality of estimators based on truncated-likelihood functions are preserved. We apply our result to the family of generalized Wendland covariance functions and discuss several examples of Wendland approximations. For families of covariance functions that are not compactly supported, we combine our results with the covariance tapering approach and show that ML estimators, based on truncated-tapered likelihood functions, asymptotically minimize the Kullback-Leibler divergence, when the taper range is fixed.

math.ST↗