Finite diagonal random matrices. (English) Zbl 1278.82027
Summary: The goal of this article is to extend some results of I. Popescu [Probab. Theory Relat. Fields 144, No. 1–2, 179–220 (2009; Zbl 1165.82012)] in several directions. We establish the limiting spectral distribution (LSD) for \(r\)-diagonal matrices under reduced moment conditions compared to those required by Popescu [loc. cit.]. We also deal with the joint convergence of several sequences of such matrices. In particular, we show that there is a large class of such matrices where the joint limit is not free while the marginals are semicircular. We also consider matrices of the form \(X_{n}X_{n}^{T}\) where \(X _{n }\) is a sequence of nonsymmetric \(r\)-diagonal random matrices and establish their limiting spectral distribution.
MSC:
82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
60B20 | Random matrices (probabilistic aspects) |
60B10 | Convergence of probability measures |
46L53 | Noncommutative probability and statistics |
46L54 | Free probability and free operator algebras |
15B52 | Random matrices (algebraic aspects) |
15A18 | Eigenvalues, singular values, and eigenvectors |
Keywords:
tridiagonal matrices; finite diagonal matrices; sample covariance type matrices; limiting spectral distribution; semicircle law; free independenceCitations:
Zbl 1165.82012References:
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