Related papers: A general version of Price's theorem

We prove Milnor-Wood inequalities for local products of manifolds. As a consequence, we establish the generalized Chern Conjecture for products $M\times \Sigma^k$ for any product of a manifold $M$ with a product of $k$ copies of a surface…

Suppose $G$ is a controllable graph of order $n$ with adjacency matrix $A$. Let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) and $\Delta=\prod_{i>j}(\alpha_i-\alpha_j)^2$ ($\alpha_i$'s are eigenvalues of $A$) be the walk matrix…

We propose a novel algorithm for the support estimation of partially known Gaussian graphical models that incorporates prior information about the underlying graph. In contrast to classical approaches that provide a point estimate based on…

In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar\'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and…

We introduce an empirical functional $\Psi$ that is an optimal uniform mean estimator: Let $F\subset L_2(\mu)$ be a class of mean zero functions, $u$ is a real valued function, and $X_1,\dots,X_N$ are independent, distributed according to…

Spectral characterizations of graphs is an important topic in spectral graph theory which has been studied extensively by researchers in recent years. The study of oriented graphs, however, has received less attention so far. In Qiu et…

For $\alpha>0$ and $\sigma > 0$, we consider the following probability distribution on $\alpha\mathbb N_0$: $\pi_{\alpha,\sigma} = \exp \big(- \frac{\sigma}{{\alpha}^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!}…

The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}|X_j|^{\alpha_j}]\geq\prod_{j=1}^{n}E[|X_j|^{\alpha_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and any non-negative real…

The norm of the $m$th derivative of the map that takes an operator to its $k$th antisymmetric tensor power is evaluated. The case $m=1$ has been studied earlier by Bhatia and Friedland [R. Bhatia and S. Friedland, Variation of Grassman…

We study the distribution of the {\it matrix product} $G_1 G_2 \cdots G_r$ of $r$ independent Gaussian matrices of various sizes, where $G_i$ is $d_{i-1} \times d_i$, and we denote $p = d_0$, $q = d_r$, and require $d_1 = d_{r-1}$. Here the…

The inductive dimension dim(G) of a finite undirected graph G=(V,E) is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the…

Gibbs-ERM learning is a natural idealized model of learning with stochastic optimization algorithms (such as Stochastic Gradient Langevin Dynamics and ---to some extent--- Stochastic Gradient Descent), while it also arises in other…

The generalized inverse Gaussian, denoted $\mathrm{GIG}(p, a, b)$, is a flexible family of distributions that includes the gamma, inverse gamma, and inverse Gaussian distributions as special cases. In addition to its applications in…

The paper deals with the fast-slow motions setups in the discrete time $X^\epsilon((n+1)\epsilon)=X^\epsilon(n\epsilon)+\epsilon B(X^\epsilon(n\epsilon),\xi(n))$, $n=0,1,...,[T/\epsilon]$ and the continuous time $\frac…

The quotient of random variables with normal distributions is examined and proven to have have power law decay, with density $f\left( x\right) \simeq f_{0}x^{-2}$, with the coefficient depending on the means and variances of the numerator…

We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$,…

Consider a sequence of Poisson random connection models (X_n,lambda_n,g_n) on R^d, where lambda_n / n^d \to lambda > 0 and g_n(x) = g(nx) for some non-increasing, integrable connection function g. Let I_n(g) be the number of isolated…

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\…

Consider a random graph $G$ of size $N$ constructed according to a \textit{graphon} $w \, : \, [0,1]^{2} \mapsto [0,1]$ as follows. First embed $N$ vertices $V = \{v_1, v_2, \ldots, v_N\}$ into the interval $[0,1]$, then for each $i < j$…

Let $X^1,\ldots, X^d$ be sigma-martingales on $(\Omega,{\cal F}, P)$. We show that every bounded martingale (with respect to the underlying filtration) admits an integral representation w.r.t. $X^1,\ldots, X^d$ if and only if there is no…