The Annals of Probability

Convergence of joint moments for independent random patterned matrices

Arup Bose, Rajat Subhra Hazra, and Koushik Saha

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It is known that the joint limit distribution of independent Wigner matrices satisfies a very special asymptotic independence, called freeness. We study the joint convergence of a few other patterned matrices, providing a framework to accommodate other joint laws. In particular, the matricial limits of symmetric circulants and reverse circulants satisfy, respectively, the classical independence and the half independence. The matricial limits of Toeplitz and Hankel matrices do not seem to submit to any easy or explicit independence/dependence notions. Their limits are not independent, free or half independent.

Article information

Ann. Probab., Volume 39, Number 4 (2011), 1607-1620.

First available in Project Euclid: 5 August 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60B10: Convergence of probability measures 46L53: Noncommutative probability and statistics 46L54: Free probability and free operator algebras

Empirical and limiting spectral distribution free algebras half commutativity half independence Hankel symmetric circulant Toeplitz and Wigner matrices noncommutative probability patterned matrices Rayleigh distribution semicircular law


Bose, Arup; Hazra, Rajat Subhra; Saha, Koushik. Convergence of joint moments for independent random patterned matrices. Ann. Probab. 39 (2011), no. 4, 1607--1620. doi:10.1214/10-AOP597.

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