Abstract
We consider a random matrix whose entries are the spatial increments of the solution to the stochastic heat equation with space-time white noise. We analyze the limit behavior of the associated Wishart matrix, by showing that it converges almost surely to a diagonal matrix (with equal diagonal terms) and the renormalized Wishart matrix satisfies a central limit theorem. Our techniques are based on the analysis on Wiener chaos, Malliavin calculus and Stein’s method.
Funding Statement
David A. C. Mollinedo was partially support by the grant of the project MATH AMSUD-CAPES/CDEFI-88887.612298/2021-00. J. Gamain and C. Tudor acknowledge partial support from the Labex CEMPI (ANR-11-LABX-007-01). C. Tudor also acknowledges partial support from the projects ANR-22-CE40-0015, MATHAMSUD (22-MATH-08), ECOS SUD (C2107), Japan Science and Technology Agency CREST JPMJCR2115 and by a grant of the Ministry of Research, Innovation and Digitalization (Romania), CNCS-UEFISCDI, PN-III-P4-PCE-2021-0921, within PNCDI III.
Acknowledgments
The authors would like to thank the anonymous referee for the constructive comments that improved the quality of this paper.
Citation
Julie Gamain. David A. C. Mollinedo. Ciprian A. Tudor. "High-dimensional regime for Wishart matrices based on the increments of the solution to the stochastic heat equation." Braz. J. Probab. Stat. 37 (2) 412 - 430, June 2023. https://doi.org/10.1214/23-BJPS574
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