On moving average representations of Banach-space valued stationary processes over LCA-groups

1980 ◽  
pp. 251-258
Author(s):  
F. Schmidt
Keyword(s):  
Banach Space ◽  
Moving Average ◽  
Stationary Processes ◽  
Lca Groups

scholarly journals SPHARMA approximations for stationary functional time series on the sphere

2021 ◽  
Author(s):  
Alessia Caponera
Keyword(s):  
Random Fields ◽  
Spectral Representation ◽  
Moving Average ◽  
Autoregressive Moving Average ◽  
Stationary Processes ◽  
Representation Theorems ◽  
Infinite Dimensional ◽  
Functional Time Series ◽  
Moving Average Processes ◽  
Natural Way

AbstractIn this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in Caponera and Marinucci (Ann Stat 49(1):346–369, 2021) and Caponera et al. (Stoch Process Appl 137:167–199, 2021); more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. Further characterizations in terms of functional spectral representation theorems and Wold-like decompositions are also established.

scholarly journals Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

2019 ◽  
Vol 23 ◽  
pp. 803-822
Author(s):  
Mikkel Slot Nielsen ◽  
Jan Pedersen
Keyword(s):  
Quadratic Form ◽  
Continuous Time ◽  
Quadratic Forms ◽  
Statistical Estimation ◽  
Limit Theorems ◽  
Stationary Process ◽  
Moving Average ◽  
Sufficient Conditions ◽  
Stationary Processes ◽  
Limiting Behavior

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper, we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.

ARIMA Algebra

2017 ◽  
Author(s):  
Richard McCleary ◽  
David McDowall ◽  
Bradley J. Bartos
Keyword(s):  
Time Series ◽  
White Noise ◽  
Moving Average ◽  
General Scheme ◽  
Stationary Processes ◽  
Arima Models ◽  
Covariance Functions ◽  
The Common ◽  
Random Shocks ◽  
Ar Models

The goal of Chapter 2 is to derive the properties of common processes and, based on these properties, to develop a general scheme for classifying processes. Stationary processes includes white noise, moving average (MA), and autoregressive (AR) processes. MA and AR models can approximate mixed ARMA models. A lag or backshift operator is used to solve ARIMA models for time series observations or random shocks. Covariance functions are derived for each of the common processes.Maximum likelihood estimates are introduced for the purposes of estimating autoregressive and moving average parameters.

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