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.

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), – ∞ <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.

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.

scholarly journals Large deviations and central limit theorems for sequential and random systems of intermittent maps

2020 ◽  
pp. 1-28
Author(s):  
MATTHEW NICOL ◽  
FELIPE PEREZ PEREIRA ◽  
ANDREW TÖRÖK
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Systems ◽  
Central Limit Theorems ◽  
Random Dynamical Systems ◽  
Stat Phys ◽  
Quenched Central Limit Theorem

Abstract We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

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