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Coupling for τ-Dependent Sequences and Applications

Author

Listed:
  • J. Dedecker

    (Laboratoire de Statistique Théorique et Appliquée)

  • C. Prieur

    (Laboratoire de Statistique et Probabilités)

Abstract

Let X be a real-valued random variable and $$M$$ a σ-algebra. We show that the minimum $${\mathbb{L}}^1$$ -distance between X and a random variable distributed as X and independant of $$M$$ can be viewed as a dependence coefficient τ( $$M$$ ,X) whose definition is comparable (but different) to that of the usual β-mixing coefficient between $$M$$ and σ(X). We compare this new coefficient to other well known measures of dependence, and we show that it can be easily computed in various situations, such as causal Bernoulli shifts or stable Markov chains defined via iterative random maps. Next, we use coupling techniques to obtain Bennett and Rosenthal-type inequalities for partial sums of τ-dependent sequences. The former is used to prove a strong invariance principle for partial sums.

Suggested Citation

  • J. Dedecker & C. Prieur, 2004. "Coupling for τ-Dependent Sequences and Applications," Journal of Theoretical Probability, Springer, vol. 17(4), pages 861-885, October.
    • Handle: RePEc:spr:jotpro:v:17:y:2004:i:4:d:10.1007_s10959-004-0578-x
    DOI: 10.1007/s10959-004-0578-x
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    References listed on IDEAS

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    1. Shao, Qi-Man, 1993. "Almost sure invariance principles for mixing sequences of random variables," Stochastic Processes and their Applications, Elsevier, vol. 48(2), pages 319-334, November.
    2. Magda Peligrad, 2001. "A Note on the Uniform Laws for Dependent Processes Via Coupling," Journal of Theoretical Probability, Springer, vol. 14(4), pages 979-988, October.
    3. Peligrad, Magda, 2002. "Some remarks on coupling of dependent random variables," Statistics & Probability Letters, Elsevier, vol. 60(2), pages 201-209, November.
    4. Jerôme Dedecker & Paul Doukhan, 2002. "A New Covariance Inequality and Applications," Working Papers 2002-25, Center for Research in Economics and Statistics.
    5. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
    6. Major, Péter, 1978. "On the invariance principle for sums of independent identically distributed random variables," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 487-517, December.
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    Cited by:

    1. Davide Giraudo, 2017. "Holderian Weak Invariance Principle for Stationary Mixing Sequences," Journal of Theoretical Probability, Springer, vol. 30(1), pages 196-211, March.
    2. Demian Pouzo, 2024. "Maximal Inequalities for Empirical Processes under General Mixing Conditions with an Application to Strong Approximations," Papers 2402.11394, arXiv.org.
    3. Xu, Haotian & Wang, Daren & Zhao, Zifeng & Yu, Yi, 2022. "Change point inference in high-dimensional regression models under temporal dependence," LIDAM Discussion Papers ISBA 2022027, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Fang Han & Yicheng Li, 2020. "Moment Bounds for Large Autocovariance Matrices Under Dependence," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1445-1492, September.

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