arXiv subjects

Arup Bose

Publications and source records attributed to Arup Bose.

At least 19 recordsLinked to original sources

arXiv2026

Convergence of random sums in non-commutative probability

Random sums of independent random variables have been extensively studied in classical probability theory. We consider random sums of self-adjoint variables from a non-commutative probability space, and establish several $*$-convergence results. In particular, we show that the joint $*$-convergence of the standardized random sum of identically distributed self-adjoint variables and the standardized stopping random variable (rv) is equivalent to the convergence of all moments of the stopping rv together with the convergence of the ratio of its mean to its variance. We obtain central limit theorems for the random sums of free, independent and half independent self-adjoint variables with both deterministic and random scaling. Furthermore, we derive some scaling $*$-convergence limits for randomly indexed self-adjoint variables.

math.PR↗

arXiv2026

Revisiting Toeplitz and Hankel random matrices via $*$-convergence of circulant-type matrices

We establish the joint $*$-convergence of a random circulant matrix and a specific deterministic diagonal matrix. We also show that the empirical spectral distributions of skew-circulant and left skew-circulant random matrices converge weakly a.s.~to complex Gaussian and symmetrized Rayleigh distributions, respectively. The $*$-convergence of symmetric Toeplitz and Hankel random matrices is well known. So is the weak convergence of their random spectrum. However, not much is known about the limits. We exploit the connections of circulant, reverse circulant, and left skew-circulants with the Hankel and Toeplitz matrices, to show the $*$-convergence of the random symmetric Toeplitz matrix to the sum of two non-commutative self-adjoint variables, each having a real Gaussian distribution. A similar result holds for the non-symmetric Toeplitz matrix, but the variables are not self-adjoint and have a complex Gaussian distribution. The random Hankel matrix is shown to converge in $*$-distribution to a sum of two self-adjoint variables, each of which has a symmetrized Rayleigh distribution. As a consequence of these results, we also obtain a different proof of the convergence of the empirical spectral distribution of symmetric Toeplitz and Hankel matrices, and a slightly different way of expressing the moments of the limit spectral distribution.

math.PR↗

arXiv2026

Eliciting core spatial association from spatial time series: a random matrix approach

Spatial time series (STS) data are fundamental to climate science, yet conventional approaches often conflate temporal co-evolution with genuine spatial dependence, obscuring subtle but critical climatic anomalies. We introduce a Random Matrix Theory (RMT)-based framework to isolate "core spatial association" by suitably trimming out strong but routine temporal signals while preserving spatial signals. Our pipeline introduces Hilbert space filling curve technique and Bergsma's correlation measure of statistical dependence, to climate modelling. Applied to the diurnal temperature range (DTR) data of India (1951-2022), the method reveals distinct spatial anomalies shaped by topography, mesoclimate, and urbanization. The approach uncovers temporal evolution in spatial dependence and demonstrates how regional climate variability is structured by both physical geography and anthropogenic influences. Beyond the Indian application, the framework is broadly applicable to diverse spatio-temporal datasets, offering a robust statistical foundation for predictive modelling, resilience planning, and policy design in the context of accelerating climate change.

stat.ME↗

arXiv2025

Patterned matrices with random walk entries

It is well known that the weak limit of a suitably scaled continuous-time random walk (CTRW) is the Brownian motion. We investigate the convergence of certain patterned random matrices whose entries are independent CTRWs and their time-changed versions, in a non-commutative probability framework. For the Wigner link function, the limits are free Brownian motion and its time-changed version driven by an inverse stable subordinator. For the symmetric circulant and the circulant with CTRW entries, we use their explicit eigenvalue expressions to define some empirical processes that converge weakly to a Brownian motion and a complex Brownian motion, respectively. For matrices with iid entries, and for elliptic matrices, the algebraic limits are equal in $*$-distribution to processes whose marginals are circular and elliptic variables, respectively. A random time-changed variant of these results is also established.

math.PR↗

arXiv2024

Spatio-temporal patterns of diurnal temperature: a random matrix approach I-case of India

We consider the spatio-temporal gridded daily diurnal temperature range (DTR) data across India during the 72-year period 1951--2022. We augment this data with information on the El Nino-Southern Oscillation (ENSO) and on the climatic regions (Stamp's and Koeppen's classification) and four seasons of India. We use various matrix theory approaches to trim out strong but routine signals, random matrix theory to remove noise, and novel empirical generalised singular-value distributions to establish retention of essential signals in the trimmed data. We make use of the spatial Bergsma statistics to measure spatial association and identify temporal change points in the spatial-association. In particular, our investigation captures a yet unknown change-point over the 72 years under study with drastic changes in spatial-association of DTR in India. It also brings out changes in spatial association with regard to ENSO. We conclude that while studying/modelling Indian DTR data, due consideration should be granted to the strong spatial association that is being persistently exhibited over decades, and provision should be kept for potential change points in the temporal behaviour, which in turn can bring moderate to dramatic changes in the spatial association pattern. Some of our analysis also reaffirms the conclusions made by other authors, regarding spatial and temporal behavior of DTR, adding our own insights. We consider the data from the yearly, seasonal and climatic zones points of view, and discover several new and interesting statistical structures which should be of interest, especially to climatologists and statisticians. Our methods are not country specific and could be used profitably for DTR data from other geographical areas.

stat.ME↗

arXiv2023

The "visible" Wigner matrix

We consider the ``visible'' Wigner matrix, a Wigner matrix whose $(i, j)$-th entry is coerced to zero if $i, j$ are co-prime. Using a recent result from elementary number theory on co-primality patterns in integers, we show that the limiting spectral distribution of this matrix exists, and give explicit descriptions of its moments in terms of infinite products over primes $p$ of certain polynomials evaluated at $1/p$. We also consider the complementary ``invisible'' Wigner matrix.

math.PR↗

arXiv2023

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

Measuring spatial association and testing spatial independence based on short time course data

Spatial association measures for univariate static spatial data are widely used. When the data is in the form of a collection of spatial vectors with the same temporal domain of interest, we construct a measure of similarity between the regions' series, using Bergsma's correlation coefficient $\rho$. Due to the special properties of $\rho$, unlike other spatial association measures which test for spatial randomness, our statistic can account for spatial pairwise independence. We have derived the asymptotic behavior of our statistic under null (independence of the regions) and alternate cases (the regions are dependent). We explore the alternate scenario of spatial dependence further, using simulations for the SAR and SMA dependence models. Finally, we provide application to modelling and testing for the presence of spatial association in COVID-19 incidence data, by using our statistic on the residuals obtained after model fitting.

stat.ME↗

arXiv2022

Assessing bivariate independence: Revisiting Bergsma's covariance

Bergsma (2006) proposed a covariance $\kappa$(X,Y) between random variables X and Y. He derived their asymptotic distributions under the null hypothesis of independence between X and Y. The non-null (dependent) case does not seem to have been studied in the literature. We derive several alternate expressions for $\kappa$. One of them leads us to a very intuitive estimator of $\kappa$(X,Y) that is a nice function of four naturally arising U-statistics. We derive the exact finite sample relation between all three estimates. The asymptotic distribution of our estimator, and hence also of the other two estimators, in the non-null (dependence) case, is then obtained by using the U-statistics central limit theorem. For specific parametric bivariate distributions, the value of $\kappa$ can be derived in terms of the natural dependence parameters of these distributions. In particular, we derive the formula for $\kappa$ when (X,Y) are distributed as Gumbel's bivariate exponential. We bring out various aspects of these estimators through extensive simulations from several prominent bivariate distributions. In particular, we investigate the empirical relationship between $\kappa$ and the dependence parameters, the distributional properties of the estimators, and the accuracy of these estimators. We also investigate the powers of these measures for testing independence, compare these among themselves, and with other well known such measures. Based on these exercises, the proposed estimator seems as good or better than its competitors both in terms of power and computing efficiency.

math.ST↗

arXiv2022

Modelling COVID-19-III: endemic spread in India

A disease in a given population is termed endemic when it exhibits a steady prevalence. We address the pertinent question as to what extent COVID-19 has turned endemic in India. There are several existing models for studying endemic behaviour, such as the extensions of the traditional temporal SIR model or the spatio-temporal endemic-epidemic model of Held et al. (2005) and its extensions. We propose a "spatio-temporal Gravity model" in a state of the art generalised linear model set up that can be deployed at various spatial resolutions. In absence of routine and quality covariates in the context of COVID-19 at finer spatial scales, we make use of extraneous covariates like air-traffic passenger count that enables us to capture the local mobility and social interactions effectively. This makes the proposed model different from the existing models. The proposed gravity model not only produces consistent estimators, but also outperforms the other models when applied to Indian COVID-19 data.

stat.AP↗

arXiv2022

$XX^T$ Matrices With Independent Entries

Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments and $p/n \to y\neq 0$, then the limiting spectral distribution (LSD) of $\frac{1}{n}S$ converges to a Mar$\check{\text{c}}$enko-Pastur law. Several extensions of this result are also known. We prove a general result on the existence of the LSD of $S$ in probability or almost surely, and in particular, many of the above results follow as special cases. At the same time several new LSD results also follow from our general result. The moments of the LSD are quite involved but can be described via a set of partitions. Unlike in the i.i.d. entries case, these partitions are not necessarily non-crossing, but are related to the special symmetric partitions which are known to appear in the LSD of (generalised) Wigner matrices with independent entries. We also investigate the existence of the LSD of $S_{A}=AA^T$ when $A$ is the $p\times n$ symmetric or the asymmetric version of any of the following four random matrices: reverse circulant, circulant, Toeplitz and Hankel. The LSD of $\frac{1}{n}S_{A}$ for the above four cases have been studied by Bose, Gangopadhyay and Sen in 2010, when the entries are i.i.d. We show that under some general assumptions on the entries of $A$, the LSD of $S_{A}$ exists and this result generalises the existing results significantly.

math.PR↗

arXiv2022

Some patterned matrices with independent entries

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread -- the $2k$th moment of the limit equals a weighted sum over different types of pair-partitions of the set $\{1, 2, \ldots, 2k\}$ and are universal. Some results are also known for the sparse case. In this paper we generalise these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.

math.PR↗

arXiv2021

Sample covariance matrices and special symmetric partitions

Suppose $X_p$ is a real $p \times n$ matrix with independent entries and consider the (unscaled) sample covariance matrix $S_p=X_pX_p^T$. The Marchenko-Pastur law was discovered as the limit of the bulk distribution of the sample covariance matrix in 1967. There have been extensions of this result in several directions. In this paper, we consider an extension that handles several of the existing ones as well as generates new results. We show that under suitable assumptions on the entries of $X_p$, the limiting spectral distribution exists in probability or almost surely. The moments are described by a set of partitions that are beyond pair partitions and non-crossing partitions and are also related to special symmetric partitions, which are known to appear in the limiting spectral distribution of Wigner-type matrices. Similar results hold for other patterned matrices such as reverse circulant, circulant, Toeplitz, and Hankel matrices.

math.PR↗

arXiv2021

Spectral measure of empirical autocovariance matrices of high dimensional Gaussian stationary processes

Consider the empirical autocovariance matrix at a given non-zero time lag based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measures in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices, at first the probabilistic behavior of the small singular values of the shifted versions of the autocovariance matrix are obtained. This is then used to infer about the large sample behaviour of the empirical spectral measure of the autocovariance matrices at any lag. Matrix orthogonal polynomials on the unit circle play a crucial role in our study.

math.ST↗

arXiv2021

Joint convergence of sample cross-covariance matrices

Suppose $X$ and $Y$ are $p\times n$ matrices each with mean $0$, variance $1$ and where all moments of any order are uniformly bounded as $p,n \to \infty$. Moreover, the entries $(X_{ij}, Y_{ij})$ are independent across $i,j$ with a common correlation $ρ$. Let $C=n^{-1}XY^*$ be the sample cross-covariance matrix. We show that if $n, p\to \infty, p/n\to y\neq 0$, then $C$ converges in the algebraic sense and the limit moments depend only on $ρ$. Independent copies of such matrices with same $p$ but different $n$, say $\{n_l\}$, different correlations $\{ρ_l\}$, and different non-zero $y$'s, say $\{y_l\}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $\sqrt{np^{-1}}(C-ρI_p)$ converges to an elliptic variable with parameter $ρ^2$. In particular, this elliptic variable is circular when $ρ=0$ and is semi-circular when $ρ=1$. If we take independent $C_l$, then the matrices $\{\sqrt{n_lp^{-1}}(C_l-ρ_l I_p)\}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.

math.PR↗

arXiv2021

Random matrices with independent entries: beyond non-crossing partitions

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, has attracted much attention. The $2k$th moment of the limit equals the number of non-crossing pair-partitions of the set $\{1, 2, \ldots, 2k\}$. There are several extensions of this result in the literature. In this paper we consider a unifying extension which also yields additional results. Suppose $W_n$ is an $n\times n$ symmetric matrix where the entries are independently distributed. We show that under suitable assumptions on the entries, the limiting spectral distribution exists in probability or almost surely. The moments of the limit can be described through a set of partitions which in general is larger than the set of non-crossing pair-partitions. This set gives rise to interesting enumerative combinatorial problems. Several existing limiting spectral distribution results follow from our results. These include results on the standard Wigner matrix, the adjacency matrix of a sparse homogeneous Erdős-Rényi graph, heavy tailed Wigner matrix, some banded Wigner matrices, and Wigner matrices with variance profile. Some new results on these models and their extensions also follow from our main results.

math.PR↗

arXiv2021

Distribution of Eigenvalues of Matrix Ensembles arising from Wigner and Palindromic Toeplitz Blocks

Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions; this correspondence has allowed RMT to successfully predict many number theoretic behaviors. However there are some operations which to date have no RMT analogue. Our motivation is to find an RMT analogue of Rankin-Selberg convolution, which constructs a new $L$-functions from an input pair. We report one such attempt; while it does not appear to model convolution, it does create new ensembles with properties hybridizing those of its constituents. For definiteness we concentrate on the ensemble of palindromic real symmetric Toeplitz (PST) matrices and the ensemble of real symmetric matrices, whose limiting spectral measures are the Gaussian and semi-circular distributions, respectively; these were chosen as they are the two extreme cases in terms of moment calculations. For a PST matrix $A$ and a real symmetric matrix $B$, we construct an ensemble of random real symmetric block matrices whose first row is $\lbrace A, B \rbrace$ and whose second row is $\lbrace B, A \rbrace$. By Markov's Method of Moments and the use of free probability, we show this ensemble converges weakly and almost surely to a new, universal distribution with a hybrid of Gaussian and semi-circular behaviors. We extend this construction by considering an iterated concatenation of matrices from an arbitrary pair of random real symmetric sub-ensembles with different limiting spectral measures. We prove that finite iterations converge to new, universal distributions with hybrid behavior, and that infinite iterations converge to the limiting spectral measure of the dominant component matrix.

math.PR↗

arXiv2021

Asymptotic freeness of sample covariance matrices via embedding

We present an alternative proof of asymptotic freeness of independent sample covariance matrices, when the dimension and the sample size grow at the same rate, by embedding these matrices into Wigner matrices of a larger order and using asymptotic freeness of independent Wigner and deterministic matrices.

math.PR↗