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On quadratic functionals of the Brownian sheet and related processes

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  • Deheuvels, Paul
  • Peccati, Giovanni
  • Yor, Marc

Abstract

Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on [0,1]2. In particular: (i) we use Fubini-type techniques to establish identities in law with quadratic functionals of other Gaussian processes, (ii) we explicitly calculate the Laplace transform of such functionals by means of Karhunen-Loève expansions, (iii) we prove central and non-central limit theorems in the spirit of Peccati and Yor [Four limit theorems involving quadratic functionals of Brownian motion and Brownian bridge, Asymptotic Methods in Stochastics, American Mathematical Society, Fields Institute Communication Series, 2004, pp. 75-87] and Nualart and Peccati [Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33(1) (2005) 177-193]. Our results extend some classical computations due to Lévy [Wiener's random function and other Laplacian random functions, in: Second Berkeley Symposium in Probability and Statistics, 1950, pp. 171-186], as well as the formulae recently obtained by Deheuvels and Martynov [Karhunen-Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, Progress in Probability, vol. 55, Birkhäuser Verlag, Basel, 2003, pp. 57-93].

Suggested Citation

  • Deheuvels, Paul & Peccati, Giovanni & Yor, Marc, 2006. "On quadratic functionals of the Brownian sheet and related processes," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 493-538, March.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:3:p:493-538
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    References listed on IDEAS

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    1. Csörgo, Miklós, 1979. "Strong approximations of the Hoeffding, Blum, Kiefer, Rosenblatt multivariate empirical process," Journal of Multivariate Analysis, Elsevier, vol. 9(1), pages 84-100, March.
    2. Einmahl, J. H. J. & Ruymgaart, F. H., 1987. "The almost sure behavior of the oscillation modulus of the multivariate empirical process," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 87-96, November.
    3. Cotterill Derek S. & Csörgö Miklós, 1985. "On The Limiting Distribution Of And Critical Values For The Hoeffding, Blum, Kiefer, Rosenblatt Independence Criterion," Statistics & Risk Modeling, De Gruyter, vol. 3(1-2), pages 1-48, February.
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    Cited by:

    1. Ivan Nourdin & David Nualart, 2010. "Central Limit Theorems for Multiple Skorokhod Integrals," Journal of Theoretical Probability, Springer, vol. 23(1), pages 39-64, March.
    2. Ćmiel, Bogdan & Ledwina, Teresa, 2020. "Validation of association," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 55-67.
    3. Giovanni Peccati & Murad S. Taqqu, 2008. "Stable Convergence of Multiple Wiener-Itô Integrals," Journal of Theoretical Probability, Springer, vol. 21(3), pages 527-570, September.
    4. Deheuvels, Paul, 2007. "A Karhunen-Loeve expansion for a mean-centered Brownian bridge," Statistics & Probability Letters, Elsevier, vol. 77(12), pages 1190-1200, July.

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