arXiv subjects
Publications and source records attributed to Arup Bose.
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.
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.
We propose an epidemiological model using an adaptive dynamic three compartment (with four states) SIR(D) model. Our approach is similar to non-parametric curve fitting in spirit and automatically adapts to key external factors, such as interventions, while retaining the parsimonious nature of the standard SIR(D) model. Initial dynamic temporal estimates of the model parameters are obtained by minimising the aggregate residual sum of squares across the number of infections, recoveries, and fatalities, over a chosen lag period. Then a geometric smoother is applied to obtain the final time series of estimates. These estimates are used to obtain dynamic temporal robust estimates of the key feature of this pandemic, namely the "reproduction number". We illustrate our method on the Indian COVID-19 data for the period March 14 - August 31, 2020. The time series data plots of the 36 states and union territories shows a clear presence of inter-regional variation in the prognosis of the epidemic. This is also bourne out by the estimates of the underlying parameters, including the reproduction numbers for the 36 regions. Due to this, an SIR(D) model, dynamic or otherwise, on the national aggregate data is not suited for robust local predictions. The time series of estimates of the model enables us to carry out daily, weekly and also long term predictions, including construction of predictive bands. We obtain an excellent agreement between the actual data and the model predicted data at the regional level. Our estimates of the current reproduction number turn out to be more than 2 in three regions (Andhra Pradesh, Maharashtra and Uttar Pradesh) and between 1.5 and 2 in 13 regions. Each of these regions have experienced an individual trajectory, which typically involves initial phase of shock(s) followed by a relatively steady lower level of the reproduction number.
Suppose $X$ is an $N \times n$ complex matrix whose entries are centered, independent, and identically distributed random variables with variance $1/n$ and whose fourth moment is of order ${\mathcal O}(n^{-2})$. In the first part of the paper, we consider the non-Hermitian matrix $X A X^* - z$, where $A$ is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and $z\neq 0$ is a complex number. Asymptotic probability bounds for the smallest singular value of this model are obtained in the large dimensional regime where $N$ and $n$ diverge to infinity at the same rate. In the second part of the paper, we consider the special case where $A = J = [1_{i-j = 1\mod n} ]$ is a circulant matrix. Using the result of the first part, it is shown that the limit eigenvalue distribution of $X J X^*$ exists in the large dimensional regime, and we determine this limit explicitly. A statistical application of this result devoted towards testing the presence of correlations within a multivariate time series is considered. Assuming that $X$ represents a ${\mathbb C}^N$-valued time series which is observed over a time window of length $n$, the matrix $X J X^*$ represents the one-step sample autocovariance matrix of this time series. Guided by the result on the limit spectral measure of this matrix, a whiteness test against an MA correlation model on the time series is introduced. Numerical simulations show the excellent performance of this test.
Let $A_m^{(1)},\ldots, A_m^{(k)}$ be $m\times m$ left-uppermost blocks of $k$ independent $n\times n$ Haar unitary matrices where $\frac{n}{m}\to α$ as $m\to \infty$, with $1<α<\infty$. Using free probability and Brown measure techniques, we find the limiting spectral distribution of $A_m^{(1)}\cdots A_m^{(k)}$.
We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.
The existence of limiting spectral distribution (LSD) of $\hatΓ_u+\hatΓ_u^*$, the symmetric sum of the sample autocovariance matrix $\hatΓ_u$ of order $u$, is known when the observations are from an infinite dimensional vector linear process with appropriate (strong) assumptions on the coefficient matrices. Under significantly weaker conditions, we prove, in a unified way, that the LSD of any symmetric polynomial in these matrices such as $\hatΓ_u+\hatΓ_u^*$, $\hatΓ_u\hatΓ_u^*$, $\hatΓ_u\hatΓ_u^*+\hatΓ_k\hatΓ_k^*$ exist. Our approach is through the more intuitive algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. All the previous results follow as special cases. We suggest statistical uses of these LSD and related results in order determination and white noise testing.
We consider an indexed class of real symmetric random matrices which generalize the symmetric Hankel and Reverse Circulant matrices. We show that the limiting spectral distributions of these matrices exist almost surely and the limit is continuous in the index. We also study other properties of the limit.
We show that the empirical spectral distribution (ESD) of the sample autocovariance matrix (ACVM) converges as the dimension increases, when the time series is a linear process with reasonable restriction on the coefficients. The limit does not depend on the distribution of the underlying driving i.i.d. sequence and its support is unbounded. This limit does not coincide with the spectral distribution of the theoretical ACVM. However, it does so if we consider a suitably tapered version of the sample ACVM. For banded sample ACVM the limit has unbounded support as long as the number of non-zero diagonals in proportion to the dimension of the matrix is bounded away from zero. If this ratio tends to zero, then the limit exists and again coincides with the spectral distribution of the theoretical ACVM. Finally, we also study the LSD of a naturally modified version of the ACVM which is not non-negative definite.
We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur-Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSDs of Schur-Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur-Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.
Limiting Spectral Distributions (LSD) of real symmetric patterned matrices have been well-studied. In this article, we consider skew-symmetric/anti-symmetric patterned random matrices and establish the LSDs of several common matrices. For the skew-symmetric Wigner, skew-symmetric Toeplitz and the skew-symmetric Circulant, the LSDs (on the imaginary axis) are the same as those in the symmetric cases. For the skew-symmetric Hankel and the skew-symmetric Reverse Circulant however, we obtain new LSDs. We also show the existence of the LSDs for the triangular versions of these matrices. We then introduce a related modification of the symmetric matrices by changing the sign of the lower triangle part of the matrices. In this case, the modified Wigner, modified Hankel and the modified Reverse Circulants have the same LSDs as their usual symmetric counterparts while new LSDs are obtained for the modified Toeplitz and the modified Symmetric Circulant.
It is known that the joint limit distribution of independent Wigner matrices satisfies a very special asymptotic independence, called freeness. We study the joint convergence of a few other patterned matrices, providing a framework to accommodate other joint laws. In particular, the matricial limits of symmetric circulants and reverse circulants satisfy, respectively, the classical independence and the half independence. The matricial limits of Toeplitz and Hankel matrices do not seem to submit to any easy or explicit independence/dependence notions. Their limits are not independent, free or half independent.
We study the joint convergence of independent copies of several patterned matrices in the noncommutative probability setup. In particular, joint convergence holds for the well known Wigner, Toeplitz, Hankel, reverse circulant and symmetric circulant matrices. We also study some properties of the limits. In particular, we show that copies of Wigner becomes asymptotically free with copies of any of the above other matrices.
We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.
This project revolves around studying estimators for parameters in different Time Series models and studying their assymptotic properties. We introduce various bootstrap techniques for the estimators obtained. Our special emphasis is on Weighted Bootstrap. We establish the consistency of this scheme in a AR model and its variations. Numerical calculations lend further support to our consistency results. Next we analyze ARCH models, and study various estimators used for different error distributions. We also present resampling techniques for estimating the distribution of the estimators. Finally by simulating data, we analyze the numerical properties of the estimators.
We study two specific symmetric random block Toeplitz (of dimension $k \times k$) matrices: where the blocks (of size $n \times n$) are (i) matrices with i.i.d. entries, and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by $\sqrt{nk}$) when (a) $k$ is fixed and $n \to\infty$ (b) $n$ is fixed and $k\rightarrow \infty$ (c) $n$ and $k$ go to $\infty$ simultaneously. Further the LSD's obtained in (a) and (b) coincide with those in (c) when $n$ or respectively $k$ tends to infinity. This limit in (c) is the semicircle law in case (i). In Case (ii) the limit is related to the limit of the random symmetric Toepiltz matrix as obtained by Bryc et al.(2006) and Hammond and Miller(2005).
Except the Toeplitz and Hankel matrices, the common patterned matrices for which the limiting spectral distribution (LSD) are known to exist, share a common property--the number of times each random variable appears in the matrix is (more or less) same across the variables. Thus it seems natural to ask what happens to the spectrum of the Toeplitz and Hankel matrices when each entry is scaled by the square root of the number of times that entry appears in the matrix instead of the uniform scaling by $n^{-1/2}$. We show that the LSD of these balanced matrices exist and derive integral formulae for the moments of the limit distribution. Curiously, it is not clear if these moments define a unique distribution.
Consider random k-circulants A_{k,n} with n tends to infinity, k=k(n) and whose input sequence \{a_l\}_{l \ge 0} is independent with mean zero and variance one and \sup_n n^{-1}\sum_{l=1}^n \E |a_l|^{2+δ}< \infty for some δ> 0. Under suitable restrictions on the sequence \{k(n)\}_{n \ge 1}, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists and identify the limits. In particular, we prove the following: Suppose g \ge 1 is fixed and p_1 is the smallest prime divisor of g. Suppose P_g=\prod_{j=1}^g E_j where \{E_j\}_{1 \le j \le g} are i.i.d. exponential random variables with mean one. (i) If k^g = -1+ s n where s=1 if g=1 and s = o(n^{p_1 -1}) if g>1, then the empirical spectral distribution of n^{-1/2}A_{k,n} converges weakly in probability to U_1P_g^{1/2g} where U_1 is uniformly distributed over the (2g)th roots of unity, independent of P_g. (ii) If g \ge 2 and k^g = 1+ s n with s = o(n^{p_1-1}) then the empirical spectral distribution of n^{-1/2}A_{k,n} converges weakly in probability to U_2P_g^{1/2g} where U_2 is uniformly distributed over the unit circle in \mathbb R^2, independent of P_g. On the other hand, if k \ge 2, k= n^{o(1)} with \gcd(n,k) = 1, and the input is i.i.d. standard normal variables, then F_{n^{-1/2}A_{k,n}} converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius r = \exp(\E [ \log \sqrt E_1]). We also show that when n=k^2+1\to \infty, and the input is i.i.d. with finite (2+δ) moment, then the spectral radius, with appropriate scaling and centering, converges to the Gumbel distribution.