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Inoue A. Explicit formulas for the inverses of Toeplitz matrices, with applications. Probab Theory Relat Fields 2022. [DOI: 10.1007/s00440-022-01162-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
AbstractWe derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the spectral density of the process. The derivation of the formulas is based on a recently developed finite prediction theory applied to the dual process of the stationary process. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of general block Toeplitz systems for a multivariate short-memory process. The second application is closed-form formulas for the inverses of truncated block Toeplitz matrices corresponding to a multivariate ARMA process. The significance of the latter is that they provide us with a linear-time algorithm to compute the solutions of corresponding block Toeplitz systems.
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Subba Rao S, Yang J. Reconciling the Gaussian and Whittle likelihood with an application to estimation in the frequency domain. Ann Stat 2021. [DOI: 10.1214/21-aos2055] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | - Junho Yang
- Institute of Statistical Science, Academia Sinica
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Affiliation(s)
| | - Han Lin Shang
- Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, Australia
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Inoue A, Kasahara Y, Pourahmadi M. Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes. BERNOULLI 2018. [DOI: 10.3150/16-bej897] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Meyer M, Jentsch C, Kreiss JP. Baxter’s inequality and sieve bootstrap for random fields. BERNOULLI 2017. [DOI: 10.3150/16-bej835] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Meyer M, McMurry T, Politis D. Baxter’s inequality for triangular arrays. MATHEMATICAL METHODS OF STATISTICS 2015. [DOI: 10.3103/s1066530715020040] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Mohammadpour M, Soltani AR. Forward Moving Average Representations for MA Processes of Finite Order: Multivariate Stationary and Periodically Correlated. COMMUN STAT-THEOR M 2013. [DOI: 10.1080/03610926.2012.656874] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Datta Gupta S, Mazumdar RR, Glynn P. On the convergence of the spectrum of finite order approximations of stationary time series. J MULTIVARIATE ANAL 2013. [DOI: 10.1016/j.jmva.2013.05.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Mohammadpour M, Soltani AR. Forward Moving Average Representation in Multivariate MA(1) Processes. COMMUN STAT-THEOR M 2010. [DOI: 10.1080/03610920902788095] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Inoue A, Kasahara Y, Phartyal P. Baxter’s inequality for fractional Brownian motion-type processes with Hurst index less than. Stat Probab Lett 2008. [DOI: 10.1016/j.spl.2008.04.014] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Inoue A, Kasahara Y. Explicit representation of finite predictor coefficients and its applications. Ann Stat 2006. [DOI: 10.1214/009053606000000209] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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