{"author":[{"last_name":"Reker","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","full_name":"Reker, Jana","first_name":"Jana"}],"_id":"18762","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"project":[{"call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"oa":1,"type":"journal_article","article_number":"191","ddc":["510"],"acknowledgement":"I am very grateful to László Erdős for suggesting the topic and many valuable discussions during my work on the project. I would also like to thank the two anonymous referees for their careful reading of the manuscript and detailed feedback.\r\nPartially supported by ERC Advanced Grants “RMTBeyond” No. 101020331 and “LDRaM” No. 884584.","has_accepted_license":"1","intvolume":"        29","article_processing_charge":"Yes","quality_controlled":"1","date_published":"2024-12-20T00:00:00Z","date_updated":"2025-01-08T08:45:06Z","file":[{"checksum":"67178feaa8630a332599d3037a3fe70e","content_type":"application/pdf","file_name":"2024_ElectrJournProbability_Reker.pdf","creator":"dernst","success":1,"date_updated":"2025-01-08T08:44:03Z","file_id":"18773","date_created":"2025-01-08T08:44:03Z","relation":"main_file","access_level":"open_access","file_size":812428}],"publication_identifier":{"eissn":["1083-6489"]},"external_id":{"arxiv":["2307.11028"]},"abstract":[{"text":"Consider the random variable $\\mathrm{Tr}( f_1(W)A_1\\dots f_k(W)A_k)$ where $W$ is an $N\\times N$ Hermitian Wigner matrix, $k\\in\\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\\dots, f_k$ as well as bounded deterministic matrices $A_1,\\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\\dots,f_k$ and the number of traceless matrices among $A_1,\\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\\geq2$. As an application, we consider the fluctuation of $\\mathrm{Tr}(\\mathrm{e}^{\\mathrm{i} tW}A_1\\mathrm{e}^{-\\mathrm{i} tW}A_2)$ around its thermal value $\\mathrm{Tr}(A_1)\\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.","lang":"eng"}],"oa_version":"Published Version","month":"12","article_type":"original","citation":{"ieee":"J. Reker, “Multi-point functional central limit theorem for Wigner matrices,” <i>Electronic Journal of Probability</i>, vol. 29. Institute of Mathematical Statistics, 2024.","short":"J. Reker, Electronic Journal of Probability 29 (2024).","chicago":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2024. <a href=\"https://doi.org/10.1214/24-EJP1247\">https://doi.org/10.1214/24-EJP1247</a>.","mla":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” <i>Electronic Journal of Probability</i>, vol. 29, 191, Institute of Mathematical Statistics, 2024, doi:<a href=\"https://doi.org/10.1214/24-EJP1247\">10.1214/24-EJP1247</a>.","ama":"Reker J. Multi-point functional central limit theorem for Wigner matrices. <i>Electronic Journal of Probability</i>. 2024;29. doi:<a href=\"https://doi.org/10.1214/24-EJP1247\">10.1214/24-EJP1247</a>","apa":"Reker, J. (2024). Multi-point functional central limit theorem for Wigner matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/24-EJP1247\">https://doi.org/10.1214/24-EJP1247</a>","ista":"Reker J. 2024. Multi-point functional central limit theorem for Wigner matrices. Electronic Journal of Probability. 29, 191."},"publication":"Electronic Journal of Probability","publisher":"Institute of Mathematical Statistics","arxiv":1,"status":"public","file_date_updated":"2025-01-08T08:44:03Z","year":"2024","date_created":"2025-01-05T23:01:58Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","publication_status":"published","related_material":{"record":[{"id":"17173","relation":"earlier_version","status":"public"}]},"day":"20","doi":"10.1214/24-EJP1247","DOAJ_listed":"1","volume":29,"OA_type":"gold","language":[{"iso":"eng"}],"title":"Multi-point functional central limit theorem for Wigner matrices","OA_place":"publisher","department":[{"_id":"LaEr"}],"corr_author":"1","ec_funded":1}