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Limiting Behavior of the Largest Eigenvalues of Random Toeplitz Matrices
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2019 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Det asymptotiska beteendet av största egenvärdet av stokastiska Toeplitz-matriser (Swedish)
Abstract [en]

We consider random symmetric Toeplitz matrices of size n. Assuming that the entries on the diagonals are independent centered random variables with finite γ-th moment (γ>2), a law of large numbers is established for the largest eigenvalue. Following the approach of Sen and Virág (2013), in the limit of large n, the largest rescaled eigenvalue is shown to converge to the limit 0.8288... . The background theory is explained and some symmetry results on the eigenvectors of the Toeplitz matrix and an auxiliary matrix are presented. A numerical investigation illustrates the rate of convergence and the oscillatory nature of the eigenvectors of the Toeplitz matrix. Finally, the possibility of proving a limiting distribution for the largest eigenvalue is discussed, and suggestions for future research are made.

Abstract [sv]

Vi betraktar stokastiska Toeplitz-matriser av storlek n. Givet att elementen på diagonalerna är oberoende, centrerade stokastiska variabler med ändligt γ-moment (γ>2), fastställer vi ett stora talens lag för det största egenvärdet. Med metoden från Sen och Virág (2013) visar vi att det största omskalade egenvärdet konvergera mot gränsen 0.8288... . Bakgrundsteorin förklaras och några symmetriresultat för Toeplitz-matrisens egenvektorer presenteras. En numerisk undersökning illustrerar konvergenshastigheten och Toeplitz-matrisens egenvektorers periodiska natur. Slutligen diskuteras möjligheten att bevisa en asymptotisk fördelning för de största egenvärderna och förslag för fortsatt forskning läggs fram.

Place, publisher, year, edition, pages
2019.
Series
TRITA-SCI-GRU ; 2019:392
Keywords [en]
Random matrix, Toeplitz matrix, largest eigenvalues, eigenvectors
Keywords [sv]
slumpmatriser, Toeplitz matrices, största egenvärde, egenvektorer
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-263763OAI: oai:DiVA.org:kth-263763DiVA, id: diva2:1373620
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2019-11-27 Created: 2019-11-27 Last updated: 2019-12-04Bibliographically approved

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