Related papers: Standard representation of multivariate functions …
In this paper, we develop a general theory on the coverage probability of random intervals defined in terms of discrete random variables with continuous parameter spaces. The theory shows that the minimum coverage probabilities of random…
The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors…
In this paper we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case.…
A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the…
Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued…
We prove that the Young measure associated with a Borel function f is a probability distribution of the random variable f(U), where U has a uniform distribution on the domain of f. As an auxiliary result, the fact that Young measures…
Bell tests are of profound statistical nature. Besides physical considerations, the proper understanding of their implications should involve detailed statistical analyses. In this regard, recent works have shown that their consequences and…
In this paper, we prove the existence of a measure-preserving bijection from unit square to unit segment. This bijection is also called the probability isomorphism between two probability spaces. Then we give a new proof of the existence of…
We give a categorical treatment, in the spirit of Baez and Fritz, of relative entropy for probability distributions defined on standard Borel spaces. We define a category suitable for reasoning about statistical inference on standard Borel…
Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity…
We study $BV$ functions in a Hilbert space $X$ endowed with a probability measure $\nu$, assuming that $\nu$ is Fomin differentiable along suitable directions. We establish basic characterizations, and we apply the general theory to…
Suppose an interval is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
A natural representation of random graphs is the random measure. The collection of product random measures, their transformations, and non-negative test functions forms a general representation of the collection of non-negative weighted…
In this article, we consider the random sampling in the image space $V$ of mixed Lebesgue space $L^{p,q}(\mathbb{R}^{n+1})$ under an idempotent integral operator. We assume some decay and regularity conditions of the kernel and approximate…
We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave…
We extend the theory of Rubio de Francia extrapolation, including off-diagonal, limited range, and $A_{\infty}$ extrapolation, to the weighted variable Lebesgue spaces. As a consequence we are able to show that a number of different…
We show that a doubling measure on the plane can give positive measure to the graph of a continuous function. This answers a question by Wang, Wen and Wen. Moreover we show that the doubling constant of the measure can be chosen to be…
Any Borel probability measure supported on a Cantor set of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs…