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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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Inoue A. Closed-form expression for finite predictor coefficients of multivariate ARMA processes. J MULTIVARIATE ANAL 2020. [DOI: 10.1016/j.jmva.2019.104578] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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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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Ing CK, Chiou HT, Guo M. Estimation of inverse autocovariance matrices for long memory processes. BERNOULLI 2016. [DOI: 10.3150/14-bej692] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Abstract
We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.
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Aspects of prediction. J Appl Probab 2016. [DOI: 10.1017/s0021900200021276] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.
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Takemura A. Exponential decay rate of partial autocorrelation coefficients of ARMA and short-memory processes. Stat Probab Lett 2016. [DOI: 10.1016/j.spl.2015.12.023] [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: 10/22/2022]
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Rupasinghe M, Samaranayake V. Asymptotic properties of sieve bootstrap prediction intervals for processes. Stat Probab Lett 2012. [DOI: 10.1016/j.spl.2012.07.011] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Bingham N. Szegö’s theorem and its probabilistic descendants. PROBABILITY SURVEYS 2012. [DOI: 10.1214/11-ps178] [Citation(s) in RCA: 30] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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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. AR and MA representation of partial autocorrelation functions, with applications. Probab Theory Relat Fields 2007. [DOI: 10.1007/s00440-007-0074-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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