Szegö’s theorem and its probabilistic descendants

Bingham, Ν. H.; London,

doi:10.1214/11-ps178

Cited by 53 publications

(74 citation statements)

References 129 publications

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“…We have in the course of the proof of Theorem 8 in fact also shown the following corollary. This condition has been extensively studied (see, for example, [44]) and is important in probability theory (work by Szegő, see, for example, [7]).…”

Section: Any Function Of This Form Is ϕ-Regularly Varying With Index ρmentioning

confidence: 99%

Beurling slow and regular variation

Bingham

Ostaszewski

2014

Trans London Maths Soc

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Section: Any Function Of This Form Is ϕ-Regularly Varying With Index ρmentioning

confidence: 99%

Beurling slow and regular variation

Bingham

Ostaszewski

2014

Trans London Maths Soc

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“…as long as the mutual information as in theorem 1 is finite, the strong Szegö theorem holds (see Bingham, 2012) and under the usual regularity conditions on the parameter space and the spectral density function, theorems 2.1 and 2.2 on chapter 2 of Dzhaparidze (1986) hold, which imply thatθ The plot highlights a neat discontinuity occurring at around p = 2, which is a reflection of the fact that the estimated model is long memory.…”

Section: Estimationmentioning

confidence: 97%

Generalised Partial Autocorrelations and the Mutual Information Between Past and Future

Luati

Proietti

2015

The Fascination of Probability, Statistics and Their Applications

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The paper introduces the generalised partial autocorrelation (GPAC) coefficients of a stationary stochastic process. The latter are related to the generalised autocovariances, the inverse Fourier transform coefficients of a power transformation of the spectral density function. By interpreting the generalised partial autocorrelations as the partial autocorrelation coefficients of an auxiliary process, we derive their properties and relate them to essential features of the original process.Based on a parameterisation suggested by Barndorff-Nielsen and Schou (1973) and on Whittle likelihood, we develop an estimation strategy for the GPAC coefficients. We further prove that the GPAC coefficients can be used to estimate the mutual information between the past and the future of a time series.

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“…in Bingham, 2012, and the references therein) in terms of generalised partial inverse autocorrelations introduced in (2.8).…”

Section: The Mutual Information Between Past and Futurementioning

confidence: 99%

Generalised Linear Cepstral Models for the Spectrum of a Time Series

Proietti¹,

Luati²

2019

STAT SINICA

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The paper introduces the class of generalised linear models with BoxCox link for the spectrum of a time series. The Box-Cox transformation of the spectral density is represented as a finite Fourier polynomial, with coefficients, that we term generalised cepstral coefficients, providing a complete characterisation of the properties of the random process. The link function depends on a power transformation parameter and encompasses the exponential model (logarithmic link), the autoregressive model (inverse link), and the moving average model (identity link). One of the merits of this model class is the possibility of nesting alternative spectral estimation methods under the same likelihood-based framework, so that the selection of a particular parametric spectrum amounts to estimating the transformation parameter. We also show that the generalised cepstral coefficients are a one to one function of the inverse partial autocorrelations of the process, which can be used to evaluate the mutual information between the past and the future of the process.

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Szegö’s theorem and its probabilistic descendants

Cited by 53 publications

References 129 publications

Beurling slow and regular variation

Beurling slow and regular variation

Generalised Partial Autocorrelations and the Mutual Information Between Past and Future

Generalised Linear Cepstral Models for the Spectrum of a Time Series

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