Asymptotics of partial sums of linear processes with changing memory parameter*

2013 ◽  
Vol 53 (2) ◽  
pp. 196-219 ◽  
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Partial Sums ◽  
Linear Processes ◽  
Memory Parameter

Linear Processes

2019 ◽  
pp. 345-381
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Asymptotic Behavior ◽  
Fourier Transforms ◽  
Gaussian Approximation ◽  
Related Result ◽  
Weak Limit ◽  
Linear Process ◽  
Partial Sums ◽  
Linear Processes ◽  
Discrete Fourier Transforms ◽  
Linear Statistics

Here we apply different methods to establish the Gaussian approximation to linear statistics of a stationary sequence, including stationary linear processes, near-stationary processes, and discrete Fourier transforms of a strictly stationary process. More precisely, we analyze the asymptotic behavior of the partial sums associated with a short-memory linear process and prove, in particular, that if a weak limit theorem holds for the partial sums of the innovations then a related result holds for the partial sums of the linear process itself. We then move to linear processes with long memory and obtain the CLT under various dependence structures for the innovations by analyzing the asymptotic behavior of linear statistics. We also deal with the invariance principle for causal linear processes or for linear statistics with weakly associated innovations. The last section deals with discrete Fourier transforms, proving, via martingale approximation, central limit behavior at almost all frequencies under almost no condition except a regularity assumption.

Introduction to Stochastic Processes

2019 ◽  
pp. 1-24
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Dynamical Systems ◽  
Stochastic Processes ◽  
Gaussian Approximation ◽  
Partial Sums ◽  
Stationary Sequences ◽  
Linear Processes ◽  
Limiting Behavior ◽  
Additive Functionals ◽  
Moderate Deviations Principle ◽  
Recursive Sequences

We start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the dependence, we introduce illustrative examples: linear processes, functions of stationary sequences, recursive sequences, dynamical systems, additive functionals of Markov chains, and self-interactions. The limiting behavior of the associated partial sums can be handled with tools developed throughout the book. We also present basic notions for stationary sequences of random variables: various definitions and constructions, and definitions of ergodicity, projective decomposition, and spectral density. Special attention is given to dynamical systems, as many of our results also apply in this context. The chapter also surveys the basic theory of the convergence of stochastic processes in distribution, and introduces the reader to tightness, finite-dimensional convergence, and the need for maximal inequalities. It ends with the concepts of the moderate deviations principle and its functional form.

scholarly journals DETECTION OF NONCONSTANT LONG MEMORY PARAMETER

2013 ◽  
Vol 29 (5) ◽  
pp. 1009-1056 ◽  
Author(s):  
Frédéric Lavancier ◽  
Remigijus Leipus ◽  
Anne Philippe ◽  
Donatas Surgailis
Keyword(s):  
Time Series ◽  
Long Memory ◽  
General Setting ◽  
Partial Sums ◽  
Gradual Change ◽  
Test Statistics ◽  
Finite Sample ◽  
Testing Procedures ◽  
Finite Sample Properties ◽  
Memory Parameter

This article deals with detection of a nonconstant long memory parameter in time series. The null hypothesis presumes stationary or nonstationary time series with a constant long memory parameter, typically an I (d) series with d > −.5 . The alternative corresponds to an increase in persistence and includes in particular an abrupt or gradual change from I (d1) to I (d2), −.5 < d1 < d2. We discuss several test statistics based on the ratio of forward and backward sample variances of the partial sums. The consistency of the tests is proved under a very general setting. We also study the behavior of these test statistics for some models with a changing memory parameter. A simulation study shows that our testing procedures have good finite sample properties and turn out to be more powerful than the KPSS-based tests (see Kwiatkowski, Phillips, Schmidt and Shin, 1992) considered in some previous works.

scholarly journals Asymptotic distribution with random indices for linear processes

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3925-3935
Author(s):  
Yu Miao ◽  
Qinghui Gao ◽  
Shuili Zhang
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Linear Process ◽  
Partial Sums ◽  
Real Numbers ◽  
Linear Processes ◽  
The Central Limit Theorem ◽  
Dependent Sequence ◽  
Random Indices

In this paper, we consider the following linear process Xn = ?? i=-? Ci?n-i, n ? Z, and establish the central limit theorem of the randomly indexed partial sums Svn := X1 +... + Xvn, where {ci,i?Z} is a sequence of real numbers, {?n,n?Z} is a stationary m-dependent sequence and {vn;n?1} is a sequence of positive integer valued random variables. In addition, in order to show the main result, we prove the central limit theorems for randomly indexed m-dependent random variables, which improve some known results.

Central limit theorems for nearly long range dependent subordinated linear processes

2020 ◽  
Vol 57 (2) ◽  
pp. 637-656
Author(s):  
Martin Wendler ◽  
Wei Biao Wu
Keyword(s):  
Long Range ◽  
Central Limit ◽  
Short Range ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Stationary Sequences ◽  
Linear Processes ◽  
Arbitrary Power

AbstractThe limit behavior of partial sums for short range dependent stationary sequences (with summable autocovariances) and for long range dependent sequences (with autocovariances summing up to infinity) differs in various aspects. We prove central limit theorems for partial sums of subordinated linear processes of arbitrary power rank which are at the border of short and long range dependence.

The structure of non-linear processes

Tellus ◽  
1973 ◽  
Vol 25 (6) ◽  
pp. 536-544 ◽  
Author(s):  
A. Quinet
Keyword(s):  
Linear Processes ◽  
Non Linear
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