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Applications of Strongly Convergent M-Estimators are discussed. Given the ubiquity of distributions across the sciences, multiple applications in the Physical, Biomedical and Social Sciences are elaborated. In one particular implementation…
As follows from the Schwartz Impossibility Theorem, multiplication of two distributions is in general impossible. Nevertheless, often one needs to multiply a distribution by a discontinuous function, not by an arbitrary distribution. In the…
In statistical applications, the normal and the Laplace distributions are often contrasted: the former as a standard tool of analysis, the latter as its robust counterpart. I discuss the convolutions of these two popular distributions and…
Using the notion of the truncated variation we obtain a new theorem on the existence and estimation of the Riemann-Stieltjes integral. As a special case of this theorem we obtain an improved version of the Lo\'{e}ve-Young inequality for the…
The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can…
In this article, we use the strong law of large numbers to give a proof of the Herschel-Maxwell theorem, which characterizes the normal distribution as the distribution of the components of a spherically symmetric random vector, provided…
We prove a generalization of van der Corput's Difference Theorem in the theory of uniform distribution by establishing a connection with unitary operators that have Lebesgue spectrum. This allows us to show, for example, that if $(x_n)_{n =…
This article investigates the asymptotic distribution of penalized estimators with non-differentiable penalties designed to recover low-dimensional pattern structures. Patterns play a central role in estimation, as they reveal the…
In this paper, we give an explicitdescription of a class of positive measures on symmetric conesdefined by their Laplace transforms in the framework of the Rieszintegrals. This work is motivated by the importance of thesemeasures in…
This paper gives an elementary proof for the following theorem: a renewal process can be represented by a doubly-stochastic Poisson process (DSPP) if and only if the Laplace-Stieltjes transform of the inter-arrival times is of the following…
Recent works have shown that the family of probability distributions with moments given by the Fuss-Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition…
We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from $s$ distributions, $p_1, p_2, \ldots, p_s$, we design testers for the…
For two subsets S and T of a given lattice L, we define a relative distributive (modular) property over L, that underlies a large family including the usual class of distributive (modular) lattices. Our proposed class will be called…
In the paper, Sturm--Liouville differential operators on time scales consisting of a finite number of isolated points and segments are considered. Such operators unify differential and difference operators. We obtain properties of their…
In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate elliptically contoured stable distributions. It is demonstrated that these distributions form a special subclass of…
The double exponential formula, or the DE formula, is a high-precision integration formula using a change of variables called a DE transformation; whereas there is a disadvantage that it is sensitive to singularities of an integrand near…
As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently…
Consider two random variables contaminated by two unknown transformations. The aim of this paper is to test the equality of those transformations. Two cases are distinguished: first, the two random variables have known distributions.…
We show that while individual Riesz transforms are two weight norm stable under biLipschitz change of variables on $A_{\infty}$ weights, they are two weight norm unstable under even rotational change of variables on doubling weights. More…
Testing hypotheses of goodness-of-fit about mixture distributions on the basis of independent but not necessarily identically distributed random vectors is considered. The hypotheses are given by a specific distribution or by a family of…