Asymptotics for Prediction Errors of Stationary Processes with Reflection Positivity

Inoue, Akihiko; Kasahara, Yukio

doi:10.1006/jmaa.2000.7101

Cited by 3 publications

(2 citation statements)

References 7 publications

(9 reference statements)

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“…Notice that the asymptotic relation (1.18), for some classes of continuous-time stationary processes that display intermediate or long memory was established in [13,18,19] and [20].…”

Section: The Prediction Problemmentioning

confidence: 98%

“…The problem of asymptotic behavior of the mean square prediction error for continuous-time stationary processes was considered by A. Arimoto [2,3], E. Hayashi [15], A. Inoue and Y. Kasahara [18,19], Y. Kasahara [20], Seghier [29]. These authors used the so-called "projecting back and forth onto infinite past and future" approach based on the Dym-Seghier formula and von Neumann's alternating projection theorem (see [8,29], and [25], Sect.…”

Section: The Prediction Problemmentioning

confidence: 99%

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Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities

Ginovyan

Mikaelyan

2008

Acta Appl Math

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The paper considers the mean square linear prediction problem for some classes of continuous-time stationary Gaussian processes with spectral densities possessing singularities. Specifically, we are interested in estimating the rate of decrease to zero of the relative prediction error of a future value of the process using the finite past, compared with the whole past, provided that the underlying process is nondeterministic and is "close" to white noise. We obtain explicit expressions and asymptotic formulae for relative prediction error in the cases where the spectral density possess either zeros (the underlying model is an anti-persistent process), or poles (the model is a long memory processes). Our approach to the problem is based on the Krein's theory of continual analogs of orthogonal polynomials and the continual analogs of Szegö theorem on Toeplitz determinants. A key fact is that the relative prediction error can be represented explicitly by means of the so-called "parameter function" which is a continual analog of the Verblunsky coefficients (or reflection parameters) associated with orthogonal polynomials on the unit circle. To this end first we discuss some properties of Krein's functions, state continual analogs of Szegö "weak" theorem, and obtain formulae for the resolvents and Fredholm determinants of the corresponding WienerHopf truncated operators.

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“…Notice that the asymptotic relation (1.18), for some classes of continuous-time stationary processes that display intermediate or long memory was established in [13,18,19] and [20].…”

Section: The Prediction Problemmentioning

confidence: 98%

Section: The Prediction Problemmentioning

confidence: 99%

Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities

Ginovyan

Mikaelyan

2008

Acta Appl Math

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Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Inoue

Kasahara

2004

Journal of Multivariate Analysis

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Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes

Inoue¹

2002

Ann. Appl. Probab.

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Asymptotics for Prediction Errors of Stationary Processes with Reflection Positivity

Abstract: We consider the stationary processes that have completely monotone autocovariance functions R( · ). We prove that regular variation of R( · ) implies an asymptotic formula for the prediction error

Cited by 3 publications

References 7 publications

Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities

Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes

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