On the optimality of McLeish's conditions for the central limit theorem

Dedecker, Jérôme

doi:10.1016/j.crma.2015.03.010

Cited by 8 publications

(10 citation statements)

References 10 publications

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“…Peligrad and Utev (2005) constructed an example showing that for any sequence of positive constants (a n ), a n → 0, there exists a stationary Markov chain such that n≥1 a n ||E(S n |ξ 0 )|| n 3/2 < ∞ but S n / √ n is not stochastically bounded. This example and other counterexamples provided by Volný (2010), Dedecker (2015) and Cuny and Lin (2016), show that, in general, condition…”

Section: Introductionsupporting

confidence: 65%

“…By the properties of the conditional expectation we see that condition (12) implies that (4) and (5) are satisfied and therefore, by point (b) of Theorem 5, the limit in (10) exists. If this limit is not 0, note that (12) cannot be satisfied.…”

Section: Proof Of Points (B) and (C) Of Theoremmentioning

confidence: 83%

“…Also, the conclusion of Corollary 7 does not hold under just (7). Dedecker (2015) constructed a relevant example, which has been reformulated in Proposition 9.5. in Cuny and Lin (2016). This example shows that there exists a Markov operator Q on some L 2 (π) and a function f ∈ L 2 0 (π) satisfying k≥1 (log k)||Q k f || 2 π < ∞ and such that ||S n || 2 /n → ∞ as n → ∞.…”

Section: Corollary 6 the Conclusion Of Theorem 5 Also Holds Formentioning

confidence: 99%

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A new CLT for additive functionals of Markov chains

Peligrad

2020

Stochastic Processes and their Applications

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In this paper we study the additive functionals of Markov chains via conditioning with respect to both past and future of the chain. We shall point out new sufficient projective conditions, which assure that the variance of partial sums of n consecutive random variables of a stationary Markov chain is linear in n. The paper also addresses the central limit theorem problem and is listing several open questions.

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Section: Introductionsupporting

confidence: 65%

Section: Proof Of Points (B) and (C) Of Theoremmentioning

confidence: 83%

Section: Corollary 6 the Conclusion Of Theorem 5 Also Holds Formentioning

confidence: 99%

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A new CLT for additive functionals of Markov chains

Peligrad

2020

Stochastic Processes and their Applications

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“…It would be interesting to know whether the condition n≥1 E 0 (X•θ n ) 2 n 1/2 < ∞ is also optimal. The optimality of the latter condition for the CLT has been recently investigated by Dedecker [16]. His arguments do not seem to apply for the LIL.…”

Section: Wip and Asip Under Projective Conditionsmentioning

confidence: 99%

Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

Cuny¹

2017

Ann. Probab.

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We prove that, for (adapted) stationary processes, the so-called Maxwell-Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of L p -valued random variables, with 1 ≤ p < ∞. In this setting we also prove the weak invariance principle, hence generalizing a result of Peligrad and Utev [45]. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.MSC 2010 subject classification: 60F17, 60F25, 60B12; Secondary: 37A50 Key words and phrases. Banach valued processes, compact law of the iterated logarithm, invariance principles, Maxwell-Woodroofe's condition.I am thankful to Jérôme Dedecker for providing him with a copy of [58]. I wish to thank two anonymous referees for valuable remarks that yielded an improved presentation of the paper.

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“…Our work is based on the paper by Hannan [13], who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and the error process. Most of short-range dependent processes satisfy the conditions on the error process, for instance the class of linear processes with summable coefficients and square integrable innovations, a large class of functions of linear processes, and many processes under various mixing conditions (see for instance [9], and also [6] for the optimality of Hannan's condition).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Distribution of Least Squares Estimators for Linear Models with Dependent Errors: Regular Designs

Caron

Dede²

2018

Math. Meth. Stat.

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In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan, who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and the error process. We show that for a large class of designs, the asymptotic covariance matrix is as simple as the independent and identically distributed case. We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions. 1 independent and identically distributed. 1 (ǫ i ) is satisfied, we propose a consistent estimator of the spectral density of (ǫ i ) (as a byproduct, we get an estimator of the covariance series).Wu and Liu [14] considered the problem of estimating the spectral density for a large class of short-range dependent processes. They proposed a consistent estimator for the spectral density, and gave some conditions under which the centered estimator satisfies a Central Limit Theorem. These results are based on the asymptotic theory of stationary processes developed by Wu [23]. This framework enables to deal with most of the statistical procedures from time series, including the estimation of the spectral density. However the class of processes satisfying the L 2 "physical dependence measure" introduced by Wu is included in the class of processes satisfying Hannan's condition. In this paper, we prove the consistency of an estimator of the spectral density of the error process under Hannan's condition. Compared to Wu's precise results on the estimation of the spectral density (Central Limit Theorem, rates of convergence, deviation inequalities), our result is only a consistency result, but it holds under Hannan's condition, that is for most of short-range dependent processes.The paper is organized as follows. In Section 2, we recall Hannan's Central Limit Theorem for the least squares estimator, and we define the class of « regular designs » (we also give many examples of such designs). In Section 3, we focus on the estimation of the spectral density of the error process under Hannan's condition. Finally, some examples of stationary processes satisfying Hannan's condition are presented in Section 4.Theorem 2.1 is very general because it includes a very large class of designs. In this paper, we will focus on the case where the design is regular in the following sense: Definition 2.3.1 (Regular design). A fixed design X is called regular if, for any j, l in {1, . . . , p}, the coefficients ρ j,l (k) do not depend on k.A large class of regular designs is the one for which the columns are regularly varying sequences. Let us recall the definition of regularly varying sequences: Definition 2.3.2 (Regularly varying sequence [21]). A sequence S(·) is regularly varying if and only if it can be written as:where −∞ < α < ∞ and L(·) is a slowly varying sequence. 2 The transpose of a matrix X is denoted by X t .

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On the optimality of McLeish's conditions for the central limit theorem

Cited by 8 publications

References 10 publications

A new CLT for additive functionals of Markov chains

A new CLT for additive functionals of Markov chains

Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

Asymptotic Distribution of Least Squares Estimators for Linear Models with Dependent Errors: Regular Designs

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