On the functional central limit theorem for stationary processes

Dedecker, Jérôme; Rio, Emmanuel

doi:10.1016/s0246-0203(00)00111-4

Cited by 139 publications

(159 citation statements)

References 16 publications

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“…Our paper joins the line of research that deals with the study of the central limit theorem under martingale like conditions. This line of results goes back to Gordin (1969) and was enriched by numerous authors including McLeish (1975, a-b), and more recently Dedecker and Rio (2000), Dedecker and Merlevède (2002), Maxwell and Woodroofe (2000) and Wu and Woodroofe (2002), among others. In this paper, we consider conditions involving the conditional expectation of sums of random variables with respect to the distant past.…”

Section: Introductionmentioning

confidence: 57%

On the Weak Invariance Principle for Stationary Sequences under Projective Criteria

Merlevède

Peligrad

2006

J Theor Probab

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In this paper we study the central limit theorem and its weak invariance principle for sums of a stationary sequence of random variables, via a martingale decomposition. Our conditions involve the conditional expectation of sums of random variables with respect to the distant past. For the sake of applications, we also give sufficient conditions involving the conditional expectation of one, or two random variables with respect to the distant past. The results are sharp and contribute to the clarification of the central limit theorem question for stationary sequences.Mathematics Subject Classifications (1991): 60 F 05, 60 F 17.

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Section: Introductionmentioning

confidence: 57%

On the Weak Invariance Principle for Stationary Sequences under Projective Criteria

Merlevède

Peligrad

2006

J Theor Probab

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“…Since (X i ) i∈Z is ergodic, we infer from Dedecker and Rio (2000) that the random variable Z n (< x, f − P (f) >) converges in distribution to a mean-zero normal distribution with variance x t Cx as soon as…”

Section: Applicationmentioning

confidence: 99%

New dependence coefficients. Examples and applications to statistics

Dedecker

Prieur

2004

Probab. Theory Relat. Fields

144

190

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To measure the dependence between a real-valued random variable X and a σ -algebra M, we consider four distances between the conditional distribution function of X given M and the distribution function of X. The coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions and dynamical systems. Starting from a new covariance inequality, we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence. We obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis, under a minimal condition on the coefficients. Using recent results, we show that our coefficients may be also used to obtain various exponential inequalities, a concentration inequality for Lipschitz functions, and a Berry-Esseen type inequality.

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“…By conditions of the theorem and Corollary 1 in Dedecker and Rio (2000), we have S n (t) σ n ⇒ W t as n → ∞.…”

Section: Applicationsmentioning

confidence: 85%