Related papers: Small ball probability for the condition number of…

For frequentist settings in which parameter randomness represents variability rather than uncertainty, the ideal measure of the support for one hypothesis over another is the difference in the posterior and prior log odds. For situations in…

In this paper, we study the condition number of a random Toeplitz matrix. Since a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic…

Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let $X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the marginals…

Let $\mu$ be a probability measure on $\text{GL}_d(\mathbb R)$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $\mu$. We study statistical properties of random variables of the…

We consider the problem of estimating the support size of a distribution $D$. Our investigations are pursued through the lens of distribution testing and seek to understand the power of conditional sampling (denoted as COND), wherein one is…

We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:$$ \partial^{\beta} u(t, x)=- \left(-\Delta\right)^{\alpha / 2} u(t, x)+ I_{0+}^{\gamma}\left[\dot{W}(t, x)\right],\quad…

Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the…

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices, where $d\geq 2$ is an integer. For any starting point $x \in \mathbb{R}_+^d$ with $|x| = 1$ and $y \in \mathbb R$, we…

Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By…

In this short note we study the asymptotic behaviour of the minima over compact intervals of Gaussian processes, whose paths are not necessarily smooth. We show that, beyond the logarithmic large deviation Gaussian estimates, this problem…

Let $a_1, \dots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$, and let $\varepsilon_1, \ldots, \varepsilon_n$ be uniformly random $\pm 1$ signs and $X = \sum_{i=1}^{n} a_i \varepsilon_i$. It is conjectured that $X = \sum_{i=1}^{n} a_i…

We prove an O(log n) bound for the expected value of the logarithm of the componentwise (and, a fortiori, the mixed) condition number of a random sparse n x n matrix. As a consequence, small bounds on the average loss of accuracy for…

In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory and inventory management. We introduce a common approach to substantially sharpen such…

We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions,…

We describe a randomized Krylov-subspace method for estimating the spectral condition number of a real matrix A or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of…

This article is a survey of the results on asymptotic behavior of small ball probabilities in $L_2$-norm. Recent progress in this field is mainly based on the methods of spectral theory of differential and integral operators.

This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum…

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability…

We consider how an agent should update her uncertainty when it is represented by a set P of probability distributions and the agent observes that a random variable X takes on value x, given that the agent makes decisions using the minimax…

The purpose of the present paper is to establish moment estimates of Rosenthal type for a rather general class of random variables satisfying certain bounds on the cumulants. We consider sequences of random variables which satisfy a central…