Random Quadratic Forms Research Articles

A truncated matrix variate gamma distribution

A truncated form of a matrix variate gamma distribution is introduced and a number of properties of this distribution such as cumulative distribution function, orthogonal invariance, moment generating function, marginal distribution of block matrices, and moments are derived. Some results on distribution of random quadratic forms are also derived.

On the distribution of equivalence classes of random symmetric p‐adic matrices

AbstractWe consider random symmetric matrices with independent entries distributed according to the Haar measure on for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over has a non‐trivial zero.

Improved Lower Bounds for Van Der Waerden Numbers

Recently, Ben Green proved that the two-color van der Waerden number ω(3, k) is bounded from below by \({k^{{b_0}\left( k \right)}}\) where \({b_0}\left( k \right) = {c_0}{\left( {{{\log \,k} \over {\log \log \,k}}} \right)^{1/3}}\). We prove a new lower bound of kb(k) with \(b\left( k \right) = {{c\log \,k} \over {\log \log \,k}}\). This is done by modifying Green’s argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result.

Asymptotic distribution of Bernoulli quadratic forms

Consider the random quadratic form Tn=∑ 1≤u<v≤nauvXuXv, where ((auv))1≤u,v≤n is a {0,1}-valued symmetric matrix with zeros on the diagonal, and X1,X2,…,Xn are i.i.d. Ber(pn), with pn∈(0,1). In this paper, we prove various characterization theorems about the limiting distribution of Tn, in the sparse regime, where pn→0 such that E(Tn)=O(1). The main result is a decomposition theorem showing that distributional limits of Tn is the sum of three components: a mixture which consists of a quadratic function of independent Poisson variables; a linear Poisson mixture, where the mean of the mixture is itself a (possibly infinite) linear combination of independent Poisson random variables; and another independent Poisson component. This is accompanied with a universality result which allows us to replace the Bernoulli distribution with a large class of other discrete distributions. Another consequence of the general theorem is a necessary and sufficient condition for Poisson convergence, where an interesting second moment phenomenon emerges.

Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices

We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy–Widom law for the largest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a central limit theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.

High-dimensional sample covariance matrices with Curie–Weiss entries

We study the limiting spectral distribution of sample covariance matrices $XX^T$, where $X$ are $p\times n$ random matrices with correlated entries, for the cases $p/n\to y\in [0,\infty)$. If $y>0$, we obtain the Mar\v{c}enko-Pastur distribution and in the case $y=0$ the semicircle distribution (after appropriate rescaling). The entries we consider are Curie-Weiss spins, which are correlated random signs, where the degree of the correlation is governed by an inverse temperature $\beta>0$. The model exhibits a phase transition at $\beta=1$. The correlation between any two entries decays at a rate of $O(np)$ for $\beta \in (0,1)$, $O(\sqrt{np}$) for $\beta=1$, and for $\beta>1$ the correlation does not vanish in the limit. In our proofs we use Stieltjes transforms and concentration of random quadratic forms.

An Alternative Matrix Skew-Normal Random Matrix and Some Properties

We propose an alternative skew-normal random matrix, which is an extension of the multivariate skew-normal vector parameterized in Vernic (A Stiint Univ Ovidius Constanta. 13, 83–96 2005, Insur. Math. Econ.38, 413–426 2006). We define the density function and then derive and apply the corresponding moment generating function to determine the mean matrix, covariance matrix, and third and fourth moments of the new skew-normal random matrix. Additionally, we derive eight marginal and two conditional density functions and provide necessary and sufficient conditions such that two pairs of sub-matrices are independent. Finally, we derive the moment generating function for a skew-normal random matrix-based quadratic form and show its relationship to the moment generating function of the noncentral Wishart and central Wishart random matrices.

Tail and moment estimates for a class of random chaoses of order two

We derive two-sided bounds for moments and tails of random quadratic forms (random chaoses of order $2$), generated by independent symmetric random variables such that $\lVert X \rVert_{2p} \leq \alpha \lVert X \rVert_p$ for any $p\geq 1$ and some $\alpha\geq 1$. Estimates are deterministic and exact up to some multiplicative constants which depend only on $\alpha$.

On random quadratic forms: supports of potential local maxima

AbstractThe selection model in population genetics is a dynamic system on the set of the probability distributions 𝒑=(p1,…,pn) of the allelesA1…,An, withpi(t+1) proportional topi(t) multiplied by ∑jfi,jpj(t), andfi,j=fj,iinterpreted as a fitness of the gene pair (Ai,Aj). It is known that 𝒑̂ is a locally stable equilibrium if and only if 𝒑̂ is a strict local maximum of the quadratic form 𝒑T𝒇𝒑. Usually, there are multiple local maxima and lim𝒑(t) depends on 𝒑(0). To address the question of atypicalbehavior of {𝒑(t)}, John Kingman considered the case when thefi,jare independent and [0,1]-uniform. He proved that with high probability (w.h.p.) no local maximum may have more than 2.49n1∕2positive components, and reduced 2.49 to 2.14 for a nonbiological case of exponentials on [0,∞). We show that the constant 2.14 serves a broad class of smooth densities on [0,1] with the increasing hazard rate. As for a lower bound, we prove that w.h.p. for allk≤2n1∕3, there are manyk-element subsets of [n] that pass a partial test to be a support of a local maximum. Still, it may well be that w.h.p. the actual supports are much smaller. In that direction, we prove that w.h.p. a support of a local maximum, which does not contain a support of a local equilibrium, is very unlikely to have size exceeding ⅔log2nand, for the uniform fitnesses, there are super-polynomially many potential supports free of local equilibriums of size close to ½log2n.

Estimating the level of affinity of a quadratic form

Abstract The level of affinity of a Boolean function is defined as the minimum number of variables such that assigning any particular values to these variables makes the function affine. The generalized level of affinity is defined as the minimum number of linear combinations of variables the values of which may be specified in such a way that the function becomes affine. For a quadratic form of rank 2r the generalized level of affinity is equal to r. We present some properties of the distribution of the rank of the random quadratic form and, as a corollary, derive an asymptotic estimate for the generalized level of affinity of quadratic forms.

The lower tail of random quadratic forms with applications to ordinary least squares

Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the “lower tail” of such a matrix, and prove that it is sub-Gaussian under a simple fourth moment assumption on the one-dimensional marginals of the random vectors. A similar result holds for more general sums of random positive semidefinite matrices, and our (relatively simple) proof uses a variant of the so-called PAC-Bayesian method for bounding empirical processes. Using this bound, we obtain a nearly optimal finite-sample result for the ordinary least squares estimator under random design.

A Robust Design for MISO Physical-Layer Multicasting Over Line-of-Sight Channels

This paper studies a robust design problem for far-field line-of-sight (LOS) channels where phase errors are present. Compared with the commonly used additive error model, the phase error model is more suitable for capturing the uncertainty in an LOS channel, as the dominant source of uncertainty lies in the phase. We consider a multiple-input single-output (MISO) multicast scenario, in which our goal is to design a beamformer that minimizes the transmit power while satisfying probabilistic signal-to-noise ratio (SNR) constraints. The probabilistic constraints give rise to a new computational challenge, as they involve random trigonometric forms. In this work, we propose to first approximate the random trigonometric form by its second-order Taylor expansion and then tackle the resulting random quadratic form using a Bernstein-type inequality. The advantage of such an approach is that an approximately optimal beamformer can be obtained using the standard semidefinite relaxation technique. In the simulations, we first show that if a non-robust design (i.e., one that does not take phase errors into account) is used, then the whole system may collapse. We then show that our proposed method is less conservative than the existing robust design based on Gaussian approximation and thus requires a lower power budget.

Random weighted projections, random quadratic forms and random eigenvectors

We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms; (2) The infinity norm of most unit eigenvectors of a random $\pm 1$ matrix is of order $O( \sqrt { \log n/n})$; (3) An estimate on the threshold for the local semi-circle law which is tight up to a $\sqrt {\log n}$ factor.

Joint CLT for several random sesquilinear forms with applications to large-dimensional spiked population models

In this paper, we derive a joint central limit theorem for random vector whose components are function of random sesquilinear forms. This result is a natural extension of the existing central limit theory on random quadratic forms. We also provide applications in random matrix theory related to large-dimensional spiked population models. For the first application, we find the joint distribution of grouped extreme sample eigenvalues correspond to the spikes. And for the second application, under the assumption that the population covariance matrix is diagonal with $k$ (fixed) simple spikes, we derive the asymptotic joint distribution of the extreme sample eigenvalue and its corresponding sample eigenvector projection.

On the trace approximations of products of Toeplitz matrices

The paper establishes error orders for integral limit approximations to the traces of products of Toeplitz matrices generated by integrable real symmetric functions defined on the unit circle. These approximations and the corresponding error bounds are of importance in the statistical analysis of discrete-time stationary processes: asymptotic distributions and large deviations of Toeplitz type random quadratic forms, estimation of the spectral parameters and functionals, etc.

Using Components of the Mahalanobis Squared Norm of the Residuals Vector for Quality Control in Local Engineering Networks

Local engineering precision networks are currently used to monitor the displacements of large dams, with safety control purposes. These one, two or three-dimensional networks are local, because they are tied to local reference frames, and they are precision networks, because they aim at millimetric, or even submillimetric, accuracy. The most important obstacles to such an accuracy are the instrumental and environmental systematic errors. The atmospheric refraction, which causes curvature of the path of the electromagnetic waves and changes its propagation velocity, is the major source of systematic errors. Effective measurement quality control strategies are of utmost importance to guarantee the quality of the results (displacements). The paper presents a quality control strategy supported by random quadratic forms that result from the decomposition of the Mahalanobis squared norm of the residuals vector. The strategy is compared to the well known data-snooping method with regard to the quality control and parameter estimation performances.

Weak convergence of the function-indexed integrated periodogram for infinite variance processes

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $\alpha$-stable innovations. Under suitable summability conditions on the series of the Fourier coefficients of the index functions, we show that the weak limits constitute $\alpha$-stable processes which have representations as infinite Fourier series with i.i.d. $\alpha$-stable coefficients. The cases $\alpha\in(0,1)$ and $\alpha\in[1,2)$ are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case $\alpha\in(0,1)$, entropy conditions are needed for $\alpha\in[1,2)$ to ensure the tightness of the sequence of integrated periodograms indexed by functions. The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Toeplitz coefficient matrices.

A Central Limit Theorem for the SINR at the LMMSE Estimator Output for Large-Dimensional Signals

This paper is devoted to the performance study of the linear minimum mean squared error (LMMSE) estimator for multidimensional signals in the large-dimension regime. Such an estimator is frequently encountered in wireless communications and in array processing, and the signal-to-interference-plus-noise ratio (SINR) at its output is a popular performance index. The SINR can be modeled as a random quadratic form which can be studied with the help of large random matrix theory, if one assumes that the dimension of the received and transmitted signals go to infinity at the same pace. This paper considers the asymptotic behavior of the SINR for a wide class of multidimensional signal models that includes general multiple-antenna as well as spread-spectrum transmission models. The expression of the deterministic approximation of the SINR in the large-dimension regime is recalled and the SINR fluctuations around this deterministic approximation are studied. These fluctuations are shown to converge in distribution to the Gaussian law in the large-dimension regime, and their variance is shown to decrease as the inverse of the signal dimension.

Central limit theorem of random quadratics forms involving random matrices

Let s i = 1 / N ( v i , 1 , … , v i , N ) T and S = ( s 1 , s 2 , … , s K ) where random variables { v ij , i , j = 1 , 2 , … , } are i.i.d. with Ev 11 4 < ∞ . The central limit theorem of the random quadratic forms s 1 T ( SS T ) i s 1 is established, which arises from the application in wireless communications.

Generalized score test of homogeneity for mixed effects models

Many important problems in psychology and biomedical studies require testing for overdispersion, correlation and heterogeneity in mixed effects and latent variable models, and score tests are particularly useful for this purpose. But the existing testing procedures depend on restrictive assumptions. In this paper we propose a class of test statistics based on a general mixed effects model to test the homogeneity hypothesis that all of the variance components are zero. Under some mild conditions, not only do we derive asymptotic distributions of the test statistics, but also propose a resampling procedure for approximating their asymptotic distributions conditional on the observed data. To overcome the technical challenge, we establish an invariance principle for random quadratic forms indexed by a parameter. A simulation study is conducted to investigate the empirical performance of the test statistics. A real data set is analyzed to illustrate the application of our theoretical results.