Fractional Brownian motion; High-dimensional regime; Malliavin calculus; Multiple stochastic integrals; Rosenblatt process; Stein’s method; Wishart matrix; Statistics and Probability
Abstract :
[en] We study the fluctuations, as d,n → ∞, of the Wishart matrix Wn,d = d1 Xn,dXn,dT associated to a n × d random matrix Xn,d with non-Gaussian entries. We analyze the limiting behavior in distribution of Wn,d in two situations: when the entries of Xn,d are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.
Disciplines :
Mathematics
Author, co-author :
Bourguin, Solesne; Department of Mathematics and Statistics, Boston University, Boston, United States
DIEZ, Charles-Philippe Manuel ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; CNRS, Laboratoire Paul Painlevé, Université de Lille, UMR 8524, Villeneuve d’Ascq, France
Tudor, Ciprian A.; CNRS, Laboratoire Paul Painlevé, Université de Lille, UMR 8524, Villeneuve d’Ascq, France
External co-authors :
yes
Language :
English
Title :
Limiting behavior of large correlated Wishart matrices with chaotic entries
Publication date :
May 2021
Journal title :
Bernoulli
ISSN :
1350-7265
eISSN :
1573-9759
Publisher :
International Statistical Institute
Volume :
27
Issue :
2
Pages :
1077 - 1102
Peer reviewed :
Peer Reviewed verified by ORBi
Funding text :
S. Bourguin was supported in part by the Simons Foundation grant 635136. C. Tudor was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and MATHAMSUD project SARC (19-MATH-06).
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