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

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.

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.

CENTRAL LIMIT THEOREMS FOR WEIGHTED SUMS OF LINEAR PROCESSES: LP -APPROXIMABILITY VERSUS BROWNIAN MOTION

2009 ◽  
Vol 25 (3) ◽  
pp. 748-763 ◽  
Author(s):  
Kairat T. Mynbaev
Keyword(s):  
Brownian Motion ◽  
Central Limit ◽  
Asymptotic Expansions ◽  
Limit Theorems ◽  
Stochastic Calculus ◽  
Central Limit Theorems ◽  
Weighted Sums ◽  
Linear Processes

Standardized slowly varying regressors are shown to be Lp-approximable. This fact allows us to provide alternative proofs of asymptotic expansions of nonstochastic quantities and central limit results due to P.C.B. Phillips, under a less stringent assumption on linear processes. The recourse to stochastic calculus related to Brownian motion can be completely dispensed with.

STUDENTIZING WEIGHTED SUMS OF LINEAR PROCESSES

2014 ◽  
Vol 35 (2) ◽  
pp. 151-172 ◽  
Author(s):  
Violetta Dalla ◽  
Liudas Giraitis ◽  
Hira L. Koul
Keyword(s):  
Weighted Sums ◽  
Linear Processes
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