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Asymptotics of random matrices and matrix valued processes
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
2001 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains three parts. In the first two papers we consider spectral properties of symmetric matrices with elements consisting of independent Ornstein Uhlenbeck processes. The eigenvalues behave as a particle system on the real line with singular interaction consisting of electrostatic repulsion and a linear restoring force. The empirical measure is known to converge weakly in a space of continuous measure valued functions.

In the first paper we let the empirical measure act on polynomial functions and on functions of the type exp(cx2). These functionals are shown to converge under suitable conditions in the space of continuous real valued functions on [0,T].

We prove in the second paper that under suitable conditions the fluctuations of the empirical measure valued process around the limiting measure valued process, appropriately scaled, converge weakly to a Gaussian distribution valued process.

In the last paper we prove limit theorems for functionals of random matrices. Assuming that <mml:math><mml:msup><mml:mi>Y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> is an <mml:math><mml:mi> n </mml:mi><mml:mo> x </mml:mo><mml:mi> n </mml:mi></mml:math> Wigner matrix we construct a new class of random matrices by letting <mml:math><mml:msup><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mo>Σ</mml:mo><mml:mi>n</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:msubsup><mml:mo>Σ</mml:mo><mml:mi>i</mml:mi><mml:mi>m</mml:mi></mml:msubsup><mml:msubsup><mml:mi>B</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:msubsup><mml:msup><mml:mi>Y</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:msubsup><mml:mi>C</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:msubsup></mml:math> where <mml:math><mml:msup><mml:mo>Σ</mml:mo><mml:mi>n</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>B</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>C</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:msubsup></mml:math> are deterministic. We study the sequence of trace functionals <mml:math><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac><mml:mi>t</mml:mi><mml:mi>r</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mi>z</mml:mi><mml:msup><mml:mi>I</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:math> and prove convergence in probability to a limit c for which a representation formula is given.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen , 2001. , 94 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 19
Keyword [en]
Mathematics, Random matrices, singular interaction, spectral distribution, fluctuation limit, distribution valued processes
Keyword [sv]
National Category
Research subject
Mathematical Statistics
URN: urn:nbn:se:uu:diva-1460ISBN: 91-506-1487-8OAI: oai:DiVA.org:uu-1460DiVA: diva2:160999
Public defence
2001-10-05, rum 247, Byggnad 2, Polacksbacken, Uppsala, 13:15
Available from: 2001-10-18 Created: 2001-10-18Bibliographically approved

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