Estimation of the mean of a stationary time series by sampling

1973 ◽  
Vol 10 (2) ◽  
pp. 419-431 ◽  
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.

Estimation of the mean of a stationary time series by sampling

1973 ◽  
Vol 10 (02) ◽  
pp. 419-431 ◽  
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), – ∞ &lt;t&lt; ∞, be a stationary time series with meancx. Let 0 &lt;τ1&lt;τ2 &lt; … &lt;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.

Asymptotic properties of certain functionals of the periodogram

1974 ◽  
Vol 11 (3) ◽  
pp. 578-581
Author(s):  
Herbert T. Davis
Keyword(s):  
Time Series ◽  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Asymptotic Properties ◽  
Innovation Process ◽  
Triangular Array ◽  
Stationary Time Series ◽  
Riemann Sums ◽  
Stationary Time ◽  
Fundamental Frequencies

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.

Asymptotic properties of certain functionals of the periodogram

1974 ◽  
Vol 11 (03) ◽  
pp. 578-581
Author(s):  
Herbert T. Davis
Keyword(s):  
Time Series ◽  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Asymptotic Properties ◽  
Innovation Process ◽  
Triangular Array ◽  
Stationary Time Series ◽  
Riemann Sums ◽  
Stationary Time ◽  
Fundamental Frequencies

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.

scholarly journals Central limit theorems for sequential and random intermittent dynamical systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1127-1153 ◽  
Author(s):  
MATTHEW NICOL ◽  
ANDREW TÖRÖK ◽  
SANDRO VAIENTI
Keyword(s):  
Time Series ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Stationary Time Series ◽  
Stationary Time

We establish self-norming central limit theorems for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau–Manneville map. We also obtain quenched central limit theorems for random compositions of these maps.

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