Central limit theorem for stationary linear processes

Peligrad, Magda; Utev, Sergey

doi:10.1214/009117906000000179

Cited by 59 publications

(73 citation statements)

References 16 publications

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“…innovations whereas Phillips and Solo (1992) developed CLT and invariance principles for sums of linear processes based on Beveridge-Nelson decomposition. Peligrad and Utev (2006) extended Ibragimov and Linnik (1971, Theorem 18.6.5) for linear processes with innovations following a more general dependence framework. Gordin (1969) introduced a general method for proving central limit theorems for stationary processes using martingale approximation.…”

Section: Introductionmentioning

confidence: 83%

Asymptotic Normality for Weighted Sums of Linear Processes

et al. 2013

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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.

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Section: Introductionmentioning

confidence: 83%

Asymptotic Normality for Weighted Sums of Linear Processes

et al. 2013

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“…However when b n → ∞, our conditions (i) and (ii) are equivalent to Wang's ones. Its proof is based on an approximation by m-dependent random fields, a coefficientaveraging procedure inspired by Peligrad and Utev [26], and a big/small blocking summation in order to apply a central limit theorem for triangular array of weighted i.i.d. random variables.…”

Section: A Central Limit Theorem For Weighted Sumsmentioning

confidence: 99%

Invariance principles for self-similar set-indexed random fields

Biermé

Durieu

2014

Trans. Amer. Math. Soc.

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Abstract. For a stationary random field (X j ) j∈Z d and some measure µ on R d , we consider the set-indexed weighted sum process Sn(A) = j∈Z d µ(nA ∩ R j ) 1 2 X j , where R j is the unit cube with lower corner j. We establish a general invariance principle under a p-stability assumption on the X j 's and an entropy condition on the class of sets A. The limit processes are selfsimilar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures µ and particular sets A, we show that these limits can be Lévy (fractional) Brownian fields or (fractional) Brownian sheets.

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“…Theorem 2.2 (c) in [20] yields a central limit theorem for strongly mixing linear triangular arrays of type (1.1). They assume that {|ξ i | 2+δ } is uniformly integrable for a certain δ > 0.…”

Section: Central Limit Theorem For Triangular Arrays Of Dependent Ranmentioning

confidence: 95%

“…In [5], the random variables ξ i are assumed to be uniformly bounded. The proof of Theorem 2.2 (c) in [20] relies on a variation on Theorem 4.1 in [25] (see Theorem B in [20]). The proof of Theorem 3.1, which is postponed to the Appendix, also makes use of a variation on Theorem 4.1 in [25] (see also [26]).…”

Section: Central Limit Theorem For Triangular Arrays Of Dependent Ranmentioning

confidence: 99%