Asymptotic properties of certain functionals of the periodogram

The asymptotic properties of the periodogram of a weakly stationary time series for the triangular array of fundamental frequencies is studied. For linear Gaussian processes, results are obtained relating the asymptotic distribution of certain Riemann sums of the periodogram of the process to those of the periodogram of the innovation process.

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Estimation of the mean of a stationary time series by sampling

Journal of Applied Probability ◽

10.1017/s0021900200095413 ◽

1973 ◽

Vol 10 (02) ◽

pp. 419-431 ◽

Cited By ~ 3

Author(s):

David R. Brillinger

Keyword(s):

Time Series ◽

Asymptotic Distribution ◽

Stationary Point ◽

Limit Theorems ◽

Stationary Time Series ◽

Stationary Time ◽

Stationary Point Process ◽

The Mean ◽

The Times ◽

Sampling Times

LetX(t), – ∞ <t< ∞, be a stationary time series with meancx. Let 0 <τ1<τ2 < … <τN≦Tdenote A given sampling times in the interval (0,T]. We determine the asymptotic distribution of the estimate [X(τ1) + … +X(τN)]/Nofcxwhen the sampling times are fixed, satisfying a form of generalised harmonic analysis requirement, and when the sampling times are the times of events of a stationary point process independent of the seriesX(t). The results obtained may be viewed as non-standard central limit theorems.

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A theory of correlation dimension for stationary time series

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1994.0095 ◽

1994 ◽

Vol 348 (1688) ◽

pp. 343-355 ◽

Cited By ~ 15

Keyword(s):

Time Series ◽

Gaussian Processes ◽

Correlation Dimension ◽

Formal Theory ◽

Stationary Time Series ◽

Correlation Integral ◽

Stationary Time ◽

Finite Dimensional ◽

Sample Correlation ◽

Local Quantity

We develop a formal theory of correlation dimension for a class of stationary time series that includes both deterministic outputs and gaussian processes with continuous paths. This theory enables us to completely analyse correlation dimension in gaussian processes via spectral methods. Our approach plus recent results on the convergence behaviour of the sample correlation integral are then used to re-examine the behaviour of gaussian power-law coloured noise. We show that the finite correlation dimension observed by Osborne & Provenzale (1989) is a local quantity entirely due to the non-ergodicity of the simulation model. We also show that non-ergodic and weakly ergodic finite-dimensional dynamical systems (such as the simple circle map) exhibit the same phenomenon.

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Estimation of the mean of a stationary time series by sampling

Journal of Applied Probability ◽

10.2307/3212358 ◽

1973 ◽

Vol 10 (2) ◽

pp. 419-431 ◽

Cited By ~ 35

Author(s):

David R. Brillinger

Keyword(s):

Time Series ◽

Asymptotic Distribution ◽

Stationary Point ◽

Limit Theorems ◽

Stationary Time Series ◽

Stationary Time ◽

Stationary Point Process ◽

The Mean ◽

The Times ◽

Sampling Times

Let X(t), – ∞ < t < ∞, be a stationary time series with mean cx. Let 0 < τ1 < τ2 < … < τN ≦ T denote A given sampling times in the interval (0, T]. We determine the asymptotic distribution of the estimate [X(τ1) + … + X(τN)]/N of cx when the sampling times are fixed, satisfying a form of generalised harmonic analysis requirement, and when the sampling times are the times of events of a stationary point process independent of the series X(t). The results obtained may be viewed as non-standard central limit theorems.

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On the spectral decomposition of stationary time series using walsh functions. II

Advances in Applied Probability ◽

10.1017/s0001867800050266 ◽

1980 ◽

Vol 12 (02) ◽

pp. 462-474 ◽

Cited By ~ 2

Author(s):

R. Kohn

Keyword(s):

Time Series ◽

Spectral Density ◽

Spectral Decomposition ◽

Asymptotic Properties ◽

Stationary Time Series ◽

Walsh Functions ◽

Stationary Time ◽

Class Of Estimators

The paper derives the asymptotic properties of a class of estimators of the Walsh–Fourier spectral density of a stationary time series. The spectral density is defined in Kohn (1980).

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On the spectral decomposition of stationary time series using walsh functions. I

Advances in Applied Probability ◽

10.1017/s0001867800033450 ◽

1980 ◽

Vol 12 (01) ◽

pp. 183-199 ◽

Cited By ~ 2

Author(s):

R. Kohn

Keyword(s):

Time Series ◽

Fourier Transform ◽

Discrete Time ◽

Spectral Decomposition ◽

Asymptotic Properties ◽

Stationary Time Series ◽

Walsh Functions ◽

Stationary Time

The paper looks at the asymptotic properties of the finite Walsh–Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.

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On the spectral decomposition of stationary time series using walsh functions. II

Advances in Applied Probability ◽

10.2307/1426606 ◽

1980 ◽

Vol 12 (2) ◽

pp. 462-474 ◽

Cited By ~ 12

Author(s):

R. Kohn

Keyword(s):

Time Series ◽

Spectral Density ◽

Spectral Decomposition ◽

Asymptotic Properties ◽

Stationary Time Series ◽

Walsh Functions ◽

Stationary Time ◽

Class Of Estimators

The paper derives the asymptotic properties of a class of estimators of the Walsh–Fourier spectral density of a stationary time series. The spectral density is defined in Kohn (1980).

Download Full-text

On the spectral decomposition of stationary time series using walsh functions. I

Advances in Applied Probability ◽

10.2307/1426501 ◽

1980 ◽

Vol 12 (1) ◽

pp. 183-199 ◽

Cited By ~ 21

Author(s):

R. Kohn

Keyword(s):

Time Series ◽

Fourier Transform ◽

Discrete Time ◽

Spectral Decomposition ◽

Asymptotic Properties ◽

Stationary Time Series ◽

Walsh Functions ◽

Stationary Time

The paper looks at the asymptotic properties of the finite Walsh–Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.

Download Full-text