IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v145y2019icp118-126.html
   My bibliography  Save this article

Stein operators for variables form the third and fourth Wiener chaoses

Author

Listed:
  • Gaunt, Robert E.

Abstract

Let Z be a standard normal random variable and let Hn denote the nth Hermite polynomial. In this note, we obtain Stein equations for the random variables H3(Z) and H4(Z), which represent a first step towards developing Stein’s method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable aZ2+bZ+c, a,b,c∈R, which leads us to a Stein equation for the non-central chi-square distribution. We also provide a discussion as to why obtaining Stein equations for Hn(Z), n≥5, is more challenging.

Suggested Citation

  • Gaunt, Robert E., 2019. "Stein operators for variables form the third and fourth Wiener chaoses," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 118-126.
    • Handle: RePEc:eee:stapro:v:145:y:2019:i:c:p:118-126
    DOI: 10.1016/j.spl.2018.09.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715218302943
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2018.09.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    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. Yoon-Tae Kim & Hyun-Suk Park, 2023. "Bound for an Approximation of Invariant Density of Diffusions via Density Formula in Malliavin Calculus," Mathematics, MDPI, vol. 11(10), pages 1-18, May.
    4. Robert E. Gaunt, 2020. "Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I," Journal of Theoretical Probability, Springer, vol. 33(1), pages 465-505, March.
    5. Peng Chen & Ivan Nourdin & Lihu Xu & Xiaochuan Yang & Rui Zhang, 2022. "Non-integrable Stable Approximation by Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1137-1186, June.
    6. Tudor, Ciprian A., 2014. "Chaos expansion and asymptotic behavior of the Pareto distribution," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 62-68.
    7. Christophe Ley & Gesine Reinert & Yvik Swan, 2014. "Approximate Computation of Expectations: the Canonical Stein Operator," Working Papers ECARES ECARES 2014-36, ULB -- Universite Libre de Bruxelles.
    8. Ley, Christophe, 2023. "When the score function is the identity function - A tale of characterizations of the normal distribution," Econometrics and Statistics, Elsevier, vol. 26(C), pages 153-160.
    9. Privault, N. & Yam, S.C.P. & Zhang, Z., 2019. "Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3376-3405.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:145:y:2019:i:c:p:118-126. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.