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Strong Approximations for a Class of Dependent Random Variables with Semi-Exponential Tails

Author

Listed:
  • Christophe Cuny

    (Univ. Brest, UMR 6205 CNRS, LMBA)

  • Jérôme Dedecker

    (Université Paris Cité, MAP5, UMR 8145 CNRS)

  • Florence Merlevède

    (Univ. Gustave Eiffel, Univ. Paris Est Créteil, LAMA, UMR 8050 CNRS)

Abstract

We give rates of convergence in the almost sure invariance principle for sums of dependent random variables with semi-exponential tails, whose coupling coefficients decrease at a sub-exponential rate. We show that the rates in the strong invariance principle are in powers of $$\log n$$ log n . We apply our results to iid products of random matrices.

Suggested Citation

  • Christophe Cuny & Jérôme Dedecker & Florence Merlevède, 2024. "Strong Approximations for a Class of Dependent Random Variables with Semi-Exponential Tails," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2234-2252, September.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:3:d:10.1007_s10959-023-01306-0
    DOI: 10.1007/s10959-023-01306-0
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    References listed on IDEAS

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    1. Cuny, Christophe & Dedecker, Jérôme & Merlevède, Florence, 2018. "An alternative to the coupling of Berkes–Liu–Wu for strong approximations," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 233-242.
    2. Liu, Quansheng & Watbled, Frédérique, 2009. "Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3101-3132, October.
    3. Cuny, Christophe & Dedecker, Jérôme & Merlevède, Florence, 2018. "On the Komlós, Major and Tusnády strong approximation for some classes of random iterates," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1347-1385.
    4. Dedecker, Jérôme & Doukhan, Paul, 2003. "A new covariance inequality and applications," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 63-80, July.
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