On quadratic functionals of the Brownian sheet and related processes

By Paul Deheuvels, Giovanni Peccati, Marc Yor. 2006

62 citations recorded by the source corpus.

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-Loeve 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 Levy [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-Loeve expansions for weighted Wiener processes and Brownian bridges via Bessel functions, Progress in Probability, vol. 55, Birkhauser Verlag, Basel, 2003, pp. 57-93].

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/55ac6c90de0774e8da66558be05f5594771c566b

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Citations in the corpus

Central Limit Theorems for Multiple Skorokhod Integrals (2010, background): “[5]), self-intersection local times of fractional Brownian motion (Hu and Nualart [7]), approximation schemes for scalar fractional differential equations (Neuenkirch and Nourdin [12]), high-frequency CLTs for random fields on homogeneous spaces (Marinucci and Peccati [10, 11] and Peccati [23]), nee”
Stein’s method on Wiener chaos (2007, background): “3 and its generalizations have been applied to a variety of frameworks, such as: p-variations of stochastic integrals with respect to Gaussian processes [2,12], quadratic functionals of bivariate Gaussian processes [14], selfintersection local times of fractional Brownian motion [24], approximation ”