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In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix…
We consider the nonparametric estimation problem of continuous probability distribution functions. For the integrated mean square error we provide the statistic corresponding to the best invariant estimator proposed by Aggarwal (1955) and…
The Haglund--Haiman--Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: $$\tilde{H}_{\mu}(X;q,t)=\sum_{\sigma: \mu\rightarrow \mathbb{P}}x^{\sigma}t^{maj(\sigma)}q^{inv(\sigma)}.$$ Inspired by…
Majorization theory is a powerful mathematical tool to compare the disorder in distributions, with wide-ranging applications in many fields including mathematics, physics, information theory, and economics. While majorization theory…
This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang \cite{Z} had revived in us an intention to produce a second edition of our book "Opera de Cribro", one…
We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables (for the strong law of…
The paper is devoted to the contribution in the Probability Theory of the well-known Soviet mathematician Alexander Yakovlevich Khintchine (1894-1959). Several of his results are described, in particular those fundamental results on the…
Recently, Agievich proposed an interesting upper bound on binomial coefficients in the de Moivre-Laplace form. In this article, we show that the latter bound, in the specific case of a central binomial coefficient, is larger than the one…
The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation,…
In literature, the central limit theorems for the product of sums of various random variables have studied. The purpose of this note is to show that this kind of results are corollary of the invariance principle.
We study the generalisation properties of majority voting on finite ensembles of classifiers, proving margin-based generalisation bounds via the PAC-Bayes theory. These provide state-of-the-art guarantees on a number of classification…
In a recent breakthrough, Teyssier [Tey20] introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques…
In this monograph, very simple axiomatic definitions to the L. Schwartz' concepts of distribution and derivative of a distribution are formulated. Beyond this, a kind of generalization of that concepts is propose and corresponding axiomatic…
Standard maximum likelihood estimation cannot be applied to discrete energy-based models in the general case because the computation of exact model probabilities is intractable. Recent research has seen the proposal of several new…
Kingman Coalescent was first proposed by Kingman [7] in population genetics to describe population's genealogical structure. Now it becomes a bench-mark model for coalescent process. Extensive studies have been conducted on Kingman…
In this article, a discrete analogue of continuous Teissier distribution is presented. Its several important distributional characteristics have been derived. The estimation of the unknown parameter has been done using the method of maximum…
Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,\mu)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $\mu$ a.e.\ $x$ is often called the shrinking target problem.…
Chang's Lemma is a widely employed result in additive combinatorics. It gives bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. Recently, Bloom (2016) presented a powerful variant of…
Estimating probabilistic deformable template models is a new approach in the fields of computer vision and probabilistic atlases in computational anatomy. A first coherent statistical framework modelling the variability as a hidden random…
We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are…