arXiv subjects

Armine Bagyan

Publications and source records attributed to Armine Bagyan.

5 recordsLinked to original sources

arXiv2026

Random Linear Modulation with Spherically Symmetric Modulators

We consider the modulation of data given by random vectors $X_n \in \mathbb{R}^{d_n}$, $n \in \mathbb{N}$. For each $X_n$, one chooses an independent modulating random vector $Ξ_n \in \mathbb{R}^{d_n}$ and forms the projection $Y_n = Ξ_n'X_n$. It is shown, under regularity conditions on $X_n$ and $Ξ_n$, that $Y_n|Ξ_n$ converges weakly in probability to a normal distribution. More broadly, the conditional joint distribution of a family of projections constructed from random samples from $X_n$ and $Ξ_n$ is shown to converge weakly to a matrix normal distribution. We derive, $via$ G. Pólya's characterization of the normal distribution, a necessary and sufficient condition on $Y_n$ for $Ξ_n$ to be normally distributed. When $Ξ_n$ has a spherically symmetric distribution we deduce, through I. J. Schoenberg's characterization of the spherically symmetric characteristic functions on Hilbert spaces, that the probability density function of $Y_n|Ξ_n$ converges pointwise in certain $p$th means to a mixture of normal densities and the rate of convergence is quantified, resulting in uniform convergence. The cumulative distribution function of $Y_n|Ξ_n$ is shown to converge uniformly in those $p$th means to the distribution function of the same mixture, and a Lipschitz property is obtained. Examples of distributions satisfying our results are provided; these include Bingham distributions on hyperspheres of random radii, uniform distributions on hyperspheres and hypercubes of random volumes, and multivariate normal distributions; and examples of such $Ξ_n$ include the multivariate $t$-, multivariate Laplace, and spherically symmetric stable distributions.

math.ST↗

arXiv2024

Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions

For positive integers $d$ and $p$ such that $d \ge p$, let $\mathbb{R}^{d \times p}$ denote the set of $d \times p$ real matrices, $I_p$ be the identity matrix of order $p$, and $V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ be the Stiefel manifold in $\mathbb{R}^{d \times p}$. Complete asymptotic expansions as $d \to \infty$ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on $V_{d,p}$. The accuracy of each truncated expansion is strictly increasing in $d$; also, for sufficiently large $d$, the accuracy is strictly increasing in $m$, the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both $d \to \infty$ and $p \to \infty$. Values of $d$ and $p$ arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend recently-obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.

math.ST↗

arXiv2023

Complete Asymptotic Expansions and the High-Dimensional Bingham Distributions

For $d \ge 2$, let $X$ be a random vector having a Bingham distribution on $\mathcal{S}^{d-1}$, the unit sphere centered at the origin in $\R^d$, and let $Σ$ denote the symmetric matrix parameter of the distribution. Let $Ψ(Σ)$ be the normalizing constant of the distribution and let $\nabla Ψ_d(Σ)$ be the matrix of first-order partial derivatives of $Ψ(Σ)$ with respect to the entries of $Σ$. We derive complete asymptotic expansions for $Ψ(Σ)$ and $\nabla Ψ_d(Σ)$, as $d \to \infty$; these expansions are obtained subject to the growth condition that $\|Σ\|$, the Frobenius norm of $Σ$, satisfies $\|Σ\| \le γ_0 d^{r/2}$ for all $d$, where $γ_0 > 0$ and $r \in [0,1)$. Consequently, we obtain for the covariance matrix of $X$ an asymptotic expansion up to terms of arbitrary degree in $Σ$. Using a range of values of $d$ that have appeared in a variety of applications of high-dimensional spherical data analysis we tabulate the bounds on the remainder terms in the expansions of $Ψ(Σ)$ and $\nabla Ψ_d(Σ)$ and we demonstrate the rapid convergence of the bounds to zero as $r$ decreases.

math.ST↗

arXiv2022

Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices

We consider random walks on the cone of $m \times m$ positive definite matrices, where the underlying random matrices have orthogonally invariant distributions on the cone and the Riemannian metric is the measure of distance on the cone. By applying results of Khare and Rajaratnam (Ann. Probab., 45 (2017), 4101--4111), we obtain inequalities of Hoffmann-Jørgensen type for such random walks on the cone. In the case of the Wishart distribution $W_m(a,I_m)$, with index parameter $a$ and matrix parameter $I_m$, the identity matrix, we derive explicit and computable bounds for each term appearing in the Hoffmann-Jørgensen inequalities.

math.PR↗

arXiv2022

A Continuous-Time Markov Chain Model for the Spread of COVID-19

Since late 2019 the novel coronavirus, also known as COVID-19, has caused a pandemic that persists. This paper shows how a continuous-time Markov chain model for the spread of COVID-19 can be used to explain, and justify to undergraduate students, strategies now being used in attempts to control the virus. The material in the paper is written at the level of students who are taking an introductory course on the theory and applications of stochastic processes.

stat.AP↗