RANDOM MATRICES BY MA MODELS AND COMPOUND FREE POISSON
LAWS
Ayako Hasegawa
Noriyoshi Sakuma
Hiroaki Yoshida
Abstract: Recently, Pfaffel and Schlemm have investigated the Mar-chenko–Pastur type limit
( and ) of the sample covariance matrix , where
is the random matrix with dependence such that each row of is given by a
certain linear process. They have also determined the limit spectral measure by giving the
functional equation for its Stieltjes transform.
In this paper, we will see that such a limit spectral measure is a compound free Poisson
law and, in the case where dependence is given by MA modeled Gaussian process, the
sample covariance matrix can be regarded as compound Wishart matrix and, hence, gives
the random matrix model for a compound free Poisson law. We will also give an
application of compound Wishart matrix to the statistical data analysis of times series.
2000 AMS Mathematics Subject Classification: Primary: 46L54; Secondary:
60B20.
Keywords and phrases: Random matrix with dependent entries, asymptotic freeness,
compound free Poisson laws, free Meixner laws.