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We will pursue a way of building up an algebraic structure that involves, in a mathematical abstract way, the well known Grassmann variables. The problem arises when we tried to understand the grassmannian polynomial expansion on the scope…
In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and real…
As a statistical measure to quantify the topological structure of the large-scale structure in the universe, the genus number is calculated for a number of non-Gaussian distributions in which the density field is characterized by a…
We show how to use the Malliavin calculus to obtain density estimates of the law of general centered random variables. In particular, under a non-degeneracy condition, we prove and use a new formula for the density of a random variable…
One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions are a special case where the density matrix is restricted to be diagonal. Density…
We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros…
This work represents a natural coalescence of two important lines of work: learning mixtures of Gaussians and algorithmic robust statistics. In particular we give the first provably robust algorithm for learning mixtures of any constant…
In this paper, we introduce a model of Brownian polymer in a continuous random environment. The asymptotic behavior of the partition function associated to this polymer measure is studied, and we are able to separate a weak and strong…
We prove finite-sample concentration and anti-concentration bounds for dimension estimation using Gaussian kernel sums. Our bounds provide explicit dependence on sample size, bandwidth, and local geometric and distributional parameters,…
This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, comprising hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent random variables. Unlike the…
Position probability distribution of a set of massive mutually exclusive particles in one dimension has been defined. Examples with a given two mutually exclusive particles system are considered. It is emphasized that quantum particles at…
Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$…
The structure of a Bayesian network includes a great deal of information about the probability distribution of the data, which is uniquely identified given some general distributional assumptions. Therefore it's important to study its…
We study the problem of estimability of means in undirected graphical Gaussian models with symmetry restrictions represented by a colored graph. Following on from previous studies, we partition the variables into sets of vertices whose…
The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the…
In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if…
Regularization is a common tool in variational inverse problems to impose assumptions on the parameters of the problem. One such assumption is sparsity, which is commonly promoted using lasso and total variation-like regularization.…
Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume $\lambda$ is…
We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure,…
Gaussian comparison inequalities provide a way of bounding probabilities relating to multivariate Gaussian random vectors in terms of probabilities of random variables with simpler correlation structures. In this paper, we establish the…