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A bound on the Wasserstein-2 distance between linear combinations of independent random variables

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  • Arras, Benjamin
  • Azmoodeh, Ehsan
  • Poly, Guillaume
  • Swan, Yvik

Abstract

We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of ℓ2(N∗). We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.

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  • Arras, Benjamin & Azmoodeh, Ehsan & Poly, Guillaume & Swan, Yvik, 2019. "A bound on the Wasserstein-2 distance between linear combinations of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2341-2375.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:7:p:2341-2375
    DOI: 10.1016/j.spa.2018.07.009
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    2. Eden, Richard & Víquez, Juan, 2015. "Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 182-216.
    3. Kusuoka, Seiichiro & Tudor, Ciprian A., 2012. "Stein’s method for invariant measures of diffusions via Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1627-1651.
    4. Cacoullos, T. & Papathanasiou, V., 1989. "Characterizations of distributions by variance bounds," Statistics & Probability Letters, Elsevier, vol. 7(5), pages 351-356, April.
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    Cited by:

    1. Serres, Jordan, 2023. "Stability of higher order eigenvalues in dimension one," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 459-484.

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