Related papers: Generalized Gaussian processes and relations with …
Gaussian processes (GPs) are pervasive in functional data analysis, machine learning, and spatial statistics for modeling complex dependencies. Modern scientific data sets are typically heterogeneous and often contain multiple known…
Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which can be viewed as a deficiency because the exact likelihood is permutation-invariant. This article takes the alternative…
We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit…
Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in…
This paper is a brief review of recent results on the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements in a given highest-weight representation of a universal enveloping algebra ${\cal…
Gaussian process is one of the most popular non-parametric Bayesian methodologies for modeling the regression problem. It is completely determined by its mean and covariance functions. And its linear property makes it relatively…
We give a dimension-independent sparsification result for suprema of centered Gaussian processes: Let $T$ be any (possibly infinite) bounded set of vectors in $\mathbb{R}^n$, and let $\{\boldsymbol{X}_t := t \cdot \boldsymbol{g} \}_{t\in…
We find the class, ${\cal{C}}_k, k \ge 0$, of all zero mean stationary Gaussian processes, $Y(t), ~t \in \reals$ with $k$ derivatives, for which \begin{equation} Z(t) \equiv (Y^{(0)}(t), Y^{(1)}(t), \ldots, Y^{(k)}(t) ), ~ t \ge 0…
In this paper, we study semiparametric inference for linear multivariate Hawkes processes, a class of point processes widely used to describe self and mutually exciting phenomena. We establish a convolution theorem giving the best limiting…
Let $\xi$ be a L\'{e}vy process and $I_\xi(t):=\int_{0}^te^{-\xi_s}\mathrm{d} s$, $t\geq 0,$ be the exponential functional of L\'{e}vy processes on deterministic horizon. Given that $\lim_{t\to \infty}\xi_t=-\infty$ we evaluate for general…
We introduce and document a class of probability distributions, called bilateral generalized inverse Gaussian (BGIG) distributions, that are obtained by convolution of two generalized inverse Gaussian distributions supported by the positive…
A GGC (Generalized Gamma Convolution) representation of Riemann's Xi-function is constructed.
The convolution product of two conjugacy classes of the unitary group $U_n$ is described by a probability distribution on the space of central measures. Relating this convolution to the quantum cohomology of Grassmannians and using recent…
We establish a formula for moments of certain random variables involving positive continuous additive functionals (PCAFs) of standard processes which have absolutely continuous transition functions and are in duality with standard processes…
The generic identification problem is to decide whether a stochastic process $(X_t)$ is a hidden Markov process and if yes to infer its parameters for all but a subset of parametrizations that form a lower-dimensional subvariety in…
The paper addresses the question whether a random functional, a map from a set $E$ into the space of real-valued measurable functions on a probability space, has a measurable version with values in ${\mathbb R}^E$. Similarly, one may ask…
In the language of random counting measures many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition…
Gaussian processes are a natural way of defining prior distributions over functions of one or more input variables. In a simple nonparametric regression problem, where such a function gives the mean of a Gaussian distribution for an…
With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of…
We solve a conjecture raised by Evans in 1991 on the characterization of the positively correlated squared Gaussian vectors. We extend this characterization from squared Gaussian vectors to permanental vectors. As side results, we obtain…