Related papers: Integrating products of quadratic forms

Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of $k$ particles in the $n$-particle system are asymptotically independent, as $n\to\infty$ with $k$ fixed or perhaps $k=o(n)$. This…

This manuscript reviews theoretical results and applications related to quadratic forms in Gaussian random variables. It summarizes definitions, canonical representations, exact and approximate distributional results, numerical inversion…

We derive sufficient conditions for an atomic measure $\sum_{\lambda \in \Lambda} m_\lambda\, \delta_\lambda,$ where $\Lambda \subset \mathbb R^n,$ $m_\lambda$ are positive integers, and $\delta_\lambda$ is the point measure at $\lambda,$…

In this study the determinant of the average quadratic error matrix is used as the measure of state estimation efficiency. This quantity is easily computable in some cases, so it gives us a reasonable tool to find optimal measurement setup…

We study estimation of (semi-)inner products between two nonparametric probability distributions, given IID samples from each distribution. These products include relatively well-studied classical $\mathcal{L}^2$ and Sobolev inner products,…

We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random…

We derive convenient uniform concentration bounds and finite sample multivariate normal approximation results for quadratic forms, then describe some applications involving variance components estimation in linear random-effects models.…

We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack…

Let $X,X_1,X_2,\ldots$ be i.i.d. ${\mathbb{R}}^d$-valued real random vectors. Assume that ${\mathbf{E}X=0}$, $\operatorname {cov}X=\mathbb{C}$, $\mathbf{E}\Vert X\Vert^2=\sigma ^2$ and that $X$ is not concentrated in a proper subspace of…

In this paper we study the set of values of quadratic form at points of a cut and project set. We will establish conditions which ensure that the set of values is dense. Our methods involve homogeneous dynamics and we will prove a orbit…

Let $\Lambda(1,n)$ be the $(1,n)$-th Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form and $d_3(n)$ denotes the triple divisor function. This paper establishes non-trivial bounds for the averages of these arithmetic functions…

A general method to combine several estimators of the same quantity is investigated. In the spirit of model and forecast averaging, the final estimator is computed as a weighted average of the initial ones, where the weights are constrained…

We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary…

Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row…

Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and…

We consider $k$ square integrable random variables $Y_1,...,Y_k$ and $k$ random (row) vectors of length $p$, $X_1,...,X_k$ such that $X_i(l)$ is square integrable for $1\le i\le k$ and $1\le l\le p$. No assumptions whatsoever are made of…

When designing a randomized experiment, one way to ensure treatment and control groups exhibit similar covariate distributions is to randomize treatment until some prespecified level of covariate balance is satisfied; this strategy is known…

We introduce a new type of quadrature, known as approximate Gaussian quadrature (AGQ) rules using {\epsilon}-quasiorthogonality, for the approximation of integrals of the form \int f(x)d \alpha(x). The measure {\alpha}(\cdot) can be…

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…

For every positive integer $n$, the quantum integer $[n]_q$ is the polynomial $[n]_q = 1 + q + q^2 + ... + q^{n-1}.$ A quadratic addition rule for quantum integers consists of sequences of polynomials $\mathcal{R}' =…