scholarly journals ASYMPTOTIC NORMALITY FOR WEIGHTED SUMS OF LINEAR PROCESSES

2013 ◽  
Vol 30 (1) ◽  
pp. 252-284 ◽  
Author(s):  
Karim M. Abadir ◽  
Walter Distaso ◽  
Liudas Giraitis ◽  
Hira L. Koul
Keyword(s):  
Asymptotic Normality ◽  
Long Memory ◽  
Moving Average ◽  
Regression Function ◽  
Linear Process ◽  
Triangular Array ◽  
Weighted Sums ◽  
Martingale Differences ◽  
Linear Processes ◽  
Arch Models

We establish asymptotic normality of weighted sums of linear processes with general triangular array weights and when the innovations in the linear process are martingale differences. The results are obtained under minimal conditions on the weights and innovations. We also obtain weak convergence of weighted partial sum processes. The results are applicable to linear processes that have short or long memory or exhibit seasonal long memory behavior. In particular, they are applicable to GARCH and ARCH(∞) models and to their squares. They are also useful in deriving asymptotic normality of kernel-type estimators of a nonparametric regression function with short or long memory moving average errors.

Asymptotic normality of sums of Hilbert space valued random elements

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Alfredas Račkauskas
Keyword(s):  
Hilbert Space ◽  
Asymptotic Normality ◽  
Separable Hilbert Space ◽  
Linear Process ◽  
Weighted Sums ◽  
Bounded Operators ◽  
Linear Processes ◽  
Summation Methods ◽  
Random Elements

Abstract We investigate the asymptotic normality of distributions of the sequence {\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}} , {n\in\mathbb{N}} , where {(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights {(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods.

Linear processes and bispectra

1980 ◽  
Vol 17 (01) ◽  
pp. 265-270 ◽  
Author(s):  
M. Rosenblatt
Keyword(s):  
Frequency Response Function ◽  
Moving Average ◽  
Random Variables ◽  
Linear Process ◽  
Linear Filter ◽  
Linear Phase ◽  
Linear Processes ◽  
Spectral Estimates ◽  
Non Gaussian ◽  
Independent Identically Distributed

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

STATIONARY ARCH MODELS: DEPENDENCE STRUCTURE AND CENTRAL LIMIT THEOREM

2000 ◽  
Vol 16 (1) ◽  
pp. 3-22 ◽  
Author(s):  
Liudas Giraitis ◽  
Piotr Kokoszka ◽  
Remigijus Leipus
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Broad Class ◽  
Moving Average ◽  
Sufficient Conditions ◽  
Dependence Structure ◽  
Martingale Differences ◽  
Arch Models ◽  
Moving Average Representation

This paper studies a broad class of nonnegative ARCH(∞) models. Sufficient conditions for the existence of a stationary solution are established and an explicit representation of the solution as a Volterra type series is found. Under our assumptions, the covariance function can decay slowly like a power function, falling just short of the long memory structure. A moving average representation in martingale differences is established, and the central limit theorem is proved.

scholarly journals Uniform asymptotic normality of weighted sums of short-memory linear processes

2020 ◽  
Vol 57 (1) ◽  
pp. 174-195
Author(s):  
Rimas Norvaiša ◽  
Alfredas Račkauskas
Keyword(s):  
Banach Space ◽  
Change Point ◽  
Linear Process ◽  
Weighted Sums ◽  
Point Model ◽  
Linear Processes ◽  
Bounded Set ◽  
Bounded Functions ◽  
Short Memory ◽  
Change Point Model

AbstractLet $X_1, X_2,\dots$ be a short-memory linear process of random variables. For $1\leq q<2$ , let ${\mathcal{F}}$ be a bounded set of real-valued functions on [0, 1] with finite q-variation. It is proved that $\{n^{-1/2}\sum_{i=1}^nX_i\,f(i/n)\colon f\in{\mathcal{F}}\}$ converges in outer distribution in the Banach space of bounded functions on ${\mathcal{F}}$ as $n\to\infty$ . Several applications to a regression model and a multiple change point model are given.

Linear processes and bispectra

1980 ◽  
Vol 17 (1) ◽  
pp. 265-270 ◽  
Author(s):  
M. Rosenblatt
Keyword(s):  
Frequency Response Function ◽  
Moving Average ◽  
Random Variables ◽  
Linear Process ◽  
Linear Filter ◽  
Linear Phase ◽  
Linear Processes ◽  
Spectral Estimates ◽  
Non Gaussian ◽  
Independent Identically Distributed

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

scholarly journals Asymptotic Normality for Weighted Sums of Linear Processes

2012 ◽  
Author(s):  
Karim M. Abadir ◽  
Walter Distaso ◽  
Liudas Giraitis ◽  
Hira Koul
Keyword(s):  
Asymptotic Normality ◽  
Weighted Sums ◽  
Linear Processes

scholarly journals On nonparametric regression for bivariate circular long-memory time series

2021 ◽  
Author(s):  
Jan Beran ◽  
Britta Steffens ◽  
Sucharita Ghosh
Keyword(s):  
Time Series ◽  
Nonparametric Regression ◽  
Long Range ◽  
Long Memory ◽  
Regression Function ◽  
Confidence Bands ◽  
Long Range Dependence ◽  
Asymptotic Results ◽  
Memory Time ◽  
Circular Time Series

AbstractWe consider nonparametric regression for bivariate circular time series with long-range dependence. Asymptotic results for circular Nadaraya–Watson estimators are derived. Due to long-range dependence, a range of asymptotically optimal bandwidths can be found where the asymptotic rate of convergence does not depend on the bandwidth. The result can be used for obtaining simple confidence bands for the regression function. The method is illustrated by an application to wind direction data.

Simple Multivariate Conditional Covariance Dynamics Using Hyperbolically Weighted Moving Averages

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Hiroyuki Kawakatsu
Keyword(s):  
Moving Average ◽  
Curse Of Dimensionality ◽  
Moving Averages ◽  
Conditional Covariance ◽  
Arch Models ◽  
Empirical Application ◽  
Proposed Model ◽  
Out Of Sample ◽  
Weight Model ◽  
Weighted Moving Average

AbstractThis paper considers a class of multivariate ARCH models with scalar weights. A new specification with hyperbolic weighted moving average (HWMA) is proposed as an analogue of the EWMA model. Despite the restrictive dynamics of a scalar weight model, the proposed model has a number of advantages that can deal with the curse of dimensionality. The empirical application illustrates that the (pseudo) out-of-sample multistep forecasts can be surprisingly more accurate than those from the DCC model.

The distributional structure of finite moving-average processes

1988 ◽  
Vol 25 (02) ◽  
pp. 313-321 ◽  
Author(s):  
ED McKenzie
Keyword(s):  
Moving Average ◽  
Markov Property ◽  
Covariance Structure ◽  
Time Reversibility ◽  
Linear Processes ◽  
Joint Distributions ◽  
Moving Average Processes ◽  
Standard Models ◽  
Non Gaussian ◽  
Analysis Of Time Series

Analysis of time-series models has, in the past, concentrated mainly on second-order properties, i.e. the covariance structure. Recent interest in non-Gaussian and non-linear processes has necessitated exploration of more general properties, even for standard models. We demonstrate that the powerful Markov property which greatly simplifies the distributional structure of finite autoregressions has an analogue in the (non-Markovian) finite moving-average processes. In fact, all the joint distributions of samples of a qth-order moving average may be constructed from only the (q + 1)th-order distribution. The usefulness of this result is illustrated by references to three areas of application: time-reversibility; asymptotic behaviour; and sums and associated point and count processes. Generalizations of the result are also considered.

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