Searcharxiv⌕ Search

arXiv subjects

Shambhu Nath Maurya

Publications and source records attributed to Shambhu Nath Maurya.

11 recordsLinked to original sources
arXiv2025

CLT for generalized patterned random matrices: a unified approach

In this paper, we derive a unified method for establishing the distributional convergence of linear eigenvalue statistics (LES) for generalized patterned random matrices. We prove that for an $N \times N$ generalized patterned random matrix with independent subexponential entries and even degree monomial test functions of degree $p_n=o(\log N/\log \log N)$, the LES converges to standard Gaussian distribution. This generalizes the CLT results on Gaussian patterned random matrices in Chatterjee(2009), Adhikari and Saha(2017). As an application, new results on LES of Toeplitz, Hankel, circulant-type matrices and block patterned random matrices for varying test functions are derived. For odd degree monomial test functions, we derive the limiting moments of LES and show that it may not converge to a Gaussian distribution.

math.PR↗
arXiv2024

Convergence of high dimensional Toeplitz and related matrices with correlated inputs

We investigate the joint convergence of independent random Toeplitz matrices with complex input entries that have a pair-correlation structure, along with deterministic Toeplitz matrices and the backward identity permutation matrix. Further, we study the joint convergence of independent generalized Toeplitz matrices along with other related matrices. The limits depend only on the correlation structure but are universal otherwise, in that they do not depend on the underlying distributions of the entries. In particular, these results provide the joint convergence of asymmetric Hankel matrices. Earlier results in the literature on the joint convergence of random symmetric Toeplitz and symmetric Hankel matrices with real entries follow as special cases.

math.PR↗
arXiv2023

Limiting spectral distribution of Toeplitz and Hankel matrices with dependent entries

This article deals with the limiting spectral distributions (LSD) of symmetric Toeplitz and Hankel matrices with dependent entries. For any fixed positive integer $m$, we consider these $n \times n$ matrices with entries $\{Y^{(m)}_j / \sqrt n \ ;{j\geq 0}\}$, where $Y^{(m)}_j= \sum_{r=-m}^{m} X_{j+r}$ and $\{X_k\}$ are i.i.d. with mean zero and variance one. We provide an explicit expression for the moment sequences of the LSDs. As a special case, this article provides an alternate proof for the LSDs of these matrices when the entries are i.i.d. with mean zero and variance one. The method is based on the moment method. The idea of proof can also be applied to other patterned random matrices, namely reverse circulant and symmetric circulant matrices.

math.PR↗
arXiv2023

Linear Eigenvalue Statistics of $XX^\prime$ matrices

This article focuses on the fluctuations of linear eigenvalue statistics of $T_{n\times p}T'_{n\times p}$, where $T_{n\times p}$ is an $n\times p$ Toeplitz matrix with real, complex or time-dependent entries. We show that as $n \rightarrow \infty$ and $p/n \rightarrow λ\in (0, \infty)$, the linear eigenvalue statistics of these matrices for polynomial test functions converge in distribution to Gaussian random variables. We also discuss the linear eigenvalue statistics of $H_{n\times p}H'_{n\times p}$, when $H_{n\times p}$ is an $n\times p$ Hankel matrix. As a result of our studies, we also derive in-probability limit and a central limit theorem type result for Schettan norm of rectangular Toeplitz matrices. To establish the results, we use method of moments.

math.PR↗
arXiv2022

A non-Gaussian limit for linear eigenvalue statistics of Hankel matrices

This article focuses on linear eigenvalue statistics of Hankel matrices with independent entries. Using the convergence of moments we show that the linear eigenvalue statistics of Hankel matrices for odd degree monomials with degree greater than or equal to three does not converge in distribution to a Gaussian random variable. This result is a departure from the known results, Liu, Sun and Wang (2012), Kumar and Maurya (2022), of linear eigenvalue statistics of Hankel matrices for even degree monomial test functions, where the limits were Gaussian random variables.

math.PR↗
arXiv2022

On limiting spectral distribution and joint convergence of some patterned random matrices

This article deals with the limiting spectral distribution and joint convergence of reverse circulant and symmetric circulant matrices with independent entries. These results are already proved in articles Bose and Sen (2008) \cite{bose_sen_LSD_EJP}, and Bose, Hazra and Saha (2011) \cite{bose_saha_patter_JC_annals}, respectively. But this article provides alternate proofs of these results which are shorter than the existing proofs. Our method is mainly based on moment method.

math.PR↗
arXiv2021

Asymptotic Behaviour of Linear eigenvalue statistics of Hankel matrices

We study linear eigenvalue statistics of band Hankel matrices with Brownian motion entries. We prove that, the centred, normalized linear eigenvalue statistics of band Hankel matrices obey a central limit theorem (CLT) type result. We also discuss the convergence of linear eigenvalue statistics of band Hankel matrices with independent entries for odd power monomials. Our method is based on trace formula, moment method and some results of process convergence.

math.PR↗
arXiv2020

Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices

Consider the $n \times n$ reverse circulant $RC_n(t)$ and symmetric circulant $SC_n(t)$ matrices with independent Brownian motion entries. We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of these matrices as $n \tends \infty$, when the test functions of the statistics are polynomials. The proofs are mainly combinatorial, based on the trace formula, method of moments and some results on process convergence.

math.PR↗