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Gaussian Approximations of Multiple Integrals
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Giovanni Peccati, Université Paris VI
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Abstract
Fix k≥1, and let I(l), l ≥ 1, be a sequence of k-dimensional vectors of multiple Wiener-Itô integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l ->+∞ , the law of I(l) is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as I(l). The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of I(l). In particular, we will not assume that the covariance matrix of I(l) is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.
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Full text: PDF
Pages: 350-364
Published on: October 13, 2007
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Bibliography
P. Baldi and D. Marinucci (2007). Some characterizations of
the spherical harmonics coefficients for isotropic random fields.
Statistics and Probability Letters 77(5), 490-496.
J.M. Corcuera, D. Nualart and J.H.C. Woerner (2006). Power
variation of some integral long memory process. Bernoulli
12(4), 713-735. MR2248234
P. Deheuvels, G. Peccati and M. Yor (2006). On quadratic
functionals of the Brownian sheet and related processes. Stochastic
Processes and their Applications 116, 493-538. MR2199561
R.M. Dudley (2003). Real Analysis and Probability.
(2nd Edition). Cambridge University Press, Cambridge. MR1932358
Y. Hu and D. Nualart (2005). Renormalized self-intersection
local time for fractional Brownian motion. The Annals of
Probabability 33(3), 948-983. MR2135309
S. Janson (1997). Gaussian Hilbert Spaces. Cambridge
University Press, Cambridge. MR1474726
D. Marinucci (2006) High-resolution asymptotics for the
angular bispectrum of spherical random fields. The Annals of
Statistics 34, 1-41. MR2275233
D. Marinucci (2007). A Central Limit Theorem and Higher Order
Results for the Angular Bispectrum. To appear in: Probability
Theory and Related Fields.
D. Marinucci and G. Peccati (2007a). High-frequency
asymptotics for subordinated stationary fields on an Abelian compact
group. To appear in: Stochastic Processes and their Applications.
D. Marinucci and G. Peccati (2007b). Group representation and
high-frequency central limit theorems for subordinated spherical
random fields. Preprint. math.PR/0706.2851v3
A. Neuenkirch and I. Nourdin (2006). Exact rate of
convergence of some approximation schemes associated to SDEs driven
by a fractional Brownian motion. To appear in: Journal of
Theoretical Probability.
D. Nualart (2006). The Malliavin Calculus and Related
Topics (2nd Edition). Springer-Verlag. Berlin-Heidelberg-New
York. MR2200233
D. Nualart and S. Ortiz-Latorre (2007). Central limit
theorems for multiple stochastic integrals and Malliavin calculus.
To appear in: Stochastic Processes and their Applications.
D. Nualart and G. Peccati (2005). Central limit theorems for
sequences of multiple stochastic integrals. The Annals of
Probability 33, 177-193. MR2118863
G. Peccati and C.A. Tudor (2005). Gaussian limits for
vector-valued multiple stochastic integrals. In: Séminaire
de Probabilités XXXVIII, LNM 1857,
247-262, Springer-Verlag. Berlin-Heidelberg-New York. MR2126978
D. Revuz and M. Yor (1999). Continuous Martingales and
Brownian Motion. Springer-Verlag Berlin-Heidelberg-New York.
MR1725357
D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskiui
(1988). Quantum theory of angular momentum.Irreducible tensors,
spherical harmonics, vector coupling coefficients, 3nj symbols
(Translated from the Russian). World
Scientific Publishing Co., Inc., Teaneck, NJ.. MR1022665
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Electronic Communications in Probability. ISSN: 1083-589X |
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