Momar Dieng, Craig A. Tracy
This is an expository account of the edge eigenvalue distributions in random matrix theory and their application in multivariate statistics. The emphasis is on the Painlevé representations of these distributions.
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We extend the construction principle of multivariate phase-type distributions to establish an analytically tractable class of heavy-tailed multivariate random variables whose marginal distributions are of Mittag-Leffler type with arbitrary index of regular variation. The construction can essentially be seen as allowing a scalar parameter to become matrix-valued. The class of distributions is shown to be dense among all multivariate positive random variables and hence provides a versatile candidate for the modelling of heavy-tailed, but tail-independent, risks in various fields of application.
We establish a Berry--Esseen bound for general multivariate nonlinear statistics by developing a new multivariate-type randomized concentration inequality. The bound is the best possible for many known statistics. As applications, Berry--Esseen bounds for M-estimators and averaged stochastic gradient descent algorithms are obtained.
Classes of multivariate and cone valued infinitely divisible Gamma distributions are introduced. Particular emphasis is put on the cone-valued case, due to the relevance of infinitely divisible distributions on the positive semi-definite matrices in applications. The cone-valued class of generalised Gamma convolutions is studied. In particular, a characterisation in terms of an Itô-Wiener integral with respect to an infinitely divisible random measure associated to the jumps of a Lévy process is established. A new example of an infinitely divisible positive definite Gamma random matrix is intr…
We provide a new and simple characterization of the multivariate generalized Laplace distribution. In particular, this result implies that the product of a Gaussian matrix with independent and identically distributed columns by an independent isotropic Gaussian vector follows a symmetric multivariate generalized Laplace distribution.
Following the recent survey by the same author of Szegö's theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).
We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time, and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations, and computer codes based on combinatorial recursions are included…
The univariate distorted distribution were introduced in risk theory to represent changes (distortions) in the expected distributions of some risks. Later they were also applied to represent distributions of order statistics, coherent systems, proportional hazard rate (PHR) and proportional reversed hazard rate (PRHR) models, etc. In this paper we extend this concept to the multivariate setup. We show that, in some cases, they are a valid alternative to the copula representations especially when the marginal distributions may not be easily handled. Several relevant examples illustrate the appl…
In this paper, we will derive the first and 2nd order Wiener chaos decomposition for the multivariate linear statistics of the determinantal point processes associated with the spectral projection kernels on the unit spheres $S^d$. We will first get a graphical representation for the cumulants of multivariate linear statistics for any determinantal point process. The main results then follow from the very precise estimates and identities regarding the spectral projection kernels and the symmetry of the spheres.
Asymptotics for non-degenerate multivariate $U$-statistics with estimated nuisance parameters under the null and local alternative hypotheses
本文将Randles关于含估计冗余参数的U-统计量渐近理论推广到多元情形，并首次给出局部备择下的渐近分布及已知参数的影响量化，为拟合优度检验提供理论基础。
We study multilevel matrix ensembles at general beta by identifying them with a class of processes defined via the branching rules for multivariate Bessel and Heckman-Opdam hypergeometric functions. For beta = 1, 2, we express the joint multilevel density of the eigenvalues of a generalized beta-Wishart matrix as a multivariate Bessel ensemble, generalizing a result of Dieker-Warren. In the null case, we prove the conjecture of Borodin-Gorin that the joint multilevel density of the beta-Jacobi ensemble is given by a principally specialized Heckman-Opdam measure.
We provide a new general theorem for multivariate normal approximation on convex sets. The theorem is formulated in terms of a multivariate extension of Stein couplings. We apply the results to a homogeneity test in dense random graphs and to prove multivariate asymptotic normality for certain doubly indexed permutation statistics.
We investigate the multivariate central limit theorem for nonlinear statistics by means of Stein's method and Slepian's smart path interpolation method. Based on certain difference operators in theory of concentration inequalities, we obtain two explicit bounds for the rate of convergence. Applications to Rademacher functionals, the runs and quadratic forms are provided as well.
Equivalence and Separation for Multivariate Matrix-Exponential and Phase-Type Distribution Classes
本文解决Bladt和Nielsen(2010)遗留的两个问题，证明了多元矩阵指数分布类MVME与Kulkarni的代数类MME*等价，并证明相位型分布中MVPH严格包含MPH*，提供了相位型分布外的示例。
In this paper, a class of multivariate matrix-exponential affine mixtures with matrix-exponential marginals is proposed. The class is shown to possess various attractive properties such as closure under size-biased Esscher transform, order statistics, residual lifetime and higher order equilibrium distributions. This allows for explicit calculations of various actuarial quantities of interest. The results are applied in a wide range of actuarial problems including multivariate risk measures, aggregate loss, large claims reinsurance, weighted premium calculations and risk capital allocation. Fu…
In a recent paper by the authors, a new approach--called the "embedding method"--was introduced, which allows to make use of exchangeable pairs for normal and multivariate normal approximation with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.
We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $α$-stable limit distributions o…
In this paper, we extend Beta distribution to 2 by 2 matrix and give the analytical formula for its moments. Our analytical formula can be used to analyze the asymptotic behavior of Beta distribution for 2 by 2 matrix.
The stability of the multivariate geometric Brownian motion as a bilinear matrix inequality problem
本文研究多元几何布朗运动原点的全局渐近稳定性，通过构造Lyapunov函数并将稳定性条件转化为BMI可行性问题，适用于多种具体模型。
In a paper that appeared in 2010, C. Tone proved a multivariate central limit theorem for some strictly stationary random fields of random vectors satisfying certain mixing conditions. The "normalization" of a given "partial sum" (or "block sum") involved matrix multiplication by a "standard -1/2 power" of its covariance matrix (a symmetric, positive definite matrix), and the limiting multivariate normal distribution had the identity matrix as its covariance matrix. The mixing assumptions in Tone's result implicitly imposed an upper bound on the ratios of the largest to the smallest eigenvalue…
Analysis of multivariate symbol statistics in primitive rational models
该研究分析了原始有理模型中多元符号统计的渐近行为，推导了均值、协方差的渐近表达，提出了大偏差原理、多元高斯极限及中等偏差结果，并利用准幂性质和Gärtner-Ellis定理进行证明。
We employ stabilization methods and second order Poincaré inequalities to establish rates of multivariate normal convergence for a large class of vectors $(H_s^{(1)},...,H_s^{(m)})$, $s \geq 1$, of statistics of marked Poisson processes on $\mathbb{R}^d$, $d \geq 2$, as the intensity parameter $s$ tends to infinity. Our results are applicable whenever the constituent functionals $H_s^{(i)}$, $i\in\{1,...,m\}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the $d_2$-, $d_3$-, and $d_{convex}$-distances. When we compare with…
This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of Stein's equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_π$, stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the …
We present a characterization of the null moments of the Complex Multivariate Normal Distribution with non-singular covariance matrix and we give closed-forms expressions for its non-null moments.
In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein's exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.
The p-variate gamma distribution in the sense of Krishnamoorthy and Parthasarathy exists for all positive integer degrees of freedom d and at least for all real values d > p-2, p > 1. For special structures of the "associated" covariance matrix it also exists for all positive d. In this paper a relation between central and non-central multivariate gamma distributions is shown, which implies the existence of the p-variate gamma distribution at least for all non-integer d greater than the integer part of (p-1)/2 without any additional assumptions for the associated covariance matrix.
In the paper, the authors introduce a matrix-parametrized generalization of the multinomial probability mass function that involves a ratio of several multivariate gamma functions. They show the logarithmic complete monotonicity of this generalization and derive new inequalities involving ratios of multivariate gamma functions.
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