Related papers: Necessary and sufficient conditions for continuity…
A method of proving local continuity of concave functions on convex set possessing the $\mu$-compactness property is presented. This method is based on a special approximation of these functions. The class of $\mu$-compact sets can be…
We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive extension for linear functionals accompanied by additional conditions that ensure the regularity of the…
Gaussian processes are ubiquitous in nature and engineering. A case in point is a class of neural networks in the infinite-width limit, whose priors correspond to Gaussian processes. Here we perturbatively extend this correspondence to…
We consider a class of Gaussian Free Fields denoted by $(g_x)_{x \in {\cal V}_N}$, where $ {\cal V}_N = \{0,1\}^N$ and $N\in \mathbb{Z}_+$. These fields are related to a general class of $N$-dimensional random walks on the hypercube, which…
In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on $\S^2$. In particular, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the…
We develop a variational approach in order to study the qualitative properties of non-autonomous parabolic equations. Based on the method of product integrals, we discuss long-time behavior, invariance properties, and ultracontractivity of…
Despite the availability of ever more data enabled through modern sensor and computer technology, it still remains an open problem to learn dynamical systems in a sample-efficient way. We propose active learning strategies that leverage…
The paper presents bounds for the distributions of suprema for a particular class of sub-Gaussian type random fields defined over spaces with anisotropic metrics. The results are applied to random fields related to stochastic heat equations…
In this paper, we show that the methods of mathematical statistical physics can be successfully applied to random fields in finite volumes. As a result, we obtain simple necessary and sufficient conditions for the existence and uniqueness…
Gaussian processes have been successful in both supervised and unsupervised machine learning tasks, but their computational complexity has constrained practical applications. We introduce a new approximation for large-scale Gaussian…
We find sufficient conditions for the existence of an exact uniform modulus continuity for the class of $q$-isotropic Gaussian random fields introduced in [8]. We apply the result to a $d$-dimensional version of the $B^{\gamma}$ Gaussian…
Lower bounds for persistence probabilities of stationary Gaussian processes in discrete time are obtained under various conditions on the spectral measure of the process. Examples are given to show that the persistence probability can decay…
The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of…
We consider Gaussian subordinated L\'evy fields (GSLFs) that arise by subordinating L\'evy processes with positive transformations of Gaussian random fields on some spatial domain $\mathcal{D}\subset \mathbb{R}^d$, $d\geq 1$. The resulting…
We investigate the complex Gaussian as well as non-Gaussian distributed random analytical and entire functions (complex entire random field) and calculate their domain of definiteness (radius of convergence) as well as some important…
Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different manifestations of…
We consider covariance parameter estimation for Gaussian processes with functional inputs. From an increasing-domain asymptotics perspective, we prove the asymptotic consistency and normality of the maximum likelihood estimator. We extend…
A method to reconstruct fields, source strengths and physical parameters based on Gaussian process regression is presented for the case where data are known to fulfill a given linear differential equation with localized sources. The…
In this paper we define contractive and nonexpansive properties for adapted stochastic processes $X_1, X_2, \ldots $ which can be used to deduce limiting properties. In general, nonexpansive processes possess finite limits while contractive…
We extend the close interplay between continued fractions, orthogonal polynomials, and Gaussian quadrature rules to several variables in a special but natural setting which we characterize in terms of moment sequences. The crucial condition…