On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution

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dc.creatorPajor, Alain
dc.creatorPastur, Leonid
dc.date2007-10-06
dc.date.accessioned2026-07-07T08:34:37Z
dc.date.available2026-07-07T08:34:37Z
dc.descriptionWe consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with log-concave probability law and i.i.d. random amplitudes $\{τ_{α}\}_{α=1}^{m}$. This is a generalization of the case of vectors uniformly distributed over the unit sphere, studied in [Marchenko-Pastur (1967)]. We prove that if $n\to \infty, m\to \infty, m/n\to c\in \lbrack 0,\infty)$ and that the empirical eigenvalue measure of $H_{n}^{(0)}$ converges weakly, then the empirical eigenvalue measure of $H_{n,m}$ converges in probability to a non-random limit, found in [Marchenko-Pastur (1967)].
dc.identifierhttps://arxiv.org/abs/0710.1346
dc.identifierhttp://arxiv.org/abs/0710.1346
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/139500
dc.subjectProbability
dc.subjectSpectral Theory
dc.titleOn the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution
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