Block circulant matrices and the spectra of multivariate stationary sequences

Marianna Bolla,Tamás Szabados,Fatma Abdelkhalek,Máté Baranyi

doi:10.1515/spma-2020-0121

Marianna Bolla, Tamás Szabados + Show 2 more

Open Access

https://doi.org/10.1515/spma-2020-0121

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Abstract

Abstract Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the first n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a financial time series.

Highlights

Let {Xt} be a weakly stationary, real-valued time series (t ∈ Z)
The complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces
Multiplying the equations in (11) by XTj from the right, and taking expectation, the solution for the matrices Bkj and Ej (k =, . . . , n; j =, . . . , k − ) can be obtained via the block Cholesky (LDL) decomposition: Cn where Cn is nd × nd positive de nite block Toeplitz matrix of Section 2, Dn is nd × nd block diagonal and contains the positive semide nite prediction error matrices E, . . . , En in its diagonal blocks, whereas Ln is nd × nd lower triangular with blocks Bkjs below its diagonal blocks which are d × d identities, so Ln is ISE S&P return index (SP) DAX FTSE NIKKEI BOVESPA European index (EU) emerging markets index (EM)

Summary

Introduction

Let {Xt} be a weakly stationary, real-valued time series (t ∈ Z). Assume that E(Xt) = , c( ) = E(Xt ) > , and the sequence of autocovariances c(h), (h = , , . . . ) is absolutely summable; obviously, c(h) = c(−h). it is known that the Toeplitz matrix Cn = [c(i − j)]ni,j= , that is the covariance matrix of the random vector (X , . . . , Xn)T, is positive de nite for all n ∈ N (the vectors are column vectors and T denotes the transposition). Abstract: Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the rst n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞.

Results

Conclusion