Asymptotic behavior of the eigenvalues of empirical covariance matrices. I. (English. Russian original) Zbl 0788.62054

Girko, V. L.

Theory Probab. Math. Stat. 44, 37-44 (1992); translation from Teor. Veroyatn. Mat. Stat., Kiev 44, 40-48 (1991).

Summary: Under the condition that \(\varlimsup_{n\to\infty} m_ n n^{-1}< 1\), where \(n\) is the number of independent vectors \(x_ 1,\dots,x_ n\), which are observations of a random \(m_ n\)-dimensional vector \(\xi\) distributed according to the normal law \(N(a,R_{m_ n})\), it is proved that \[ p \text{-}\lim_{n\to\infty} (\mu_ 1(\widehat R)- \alpha_ 1)=0,\quad p \text{-}\lim_{n\to\infty} (\mu_{m_ n}(\widehat R)- \alpha_ 2)= 0, \] where \(\mu_{m_ n}\leq\cdots \leq \mu_ 1< c<\infty\), \(\mu_ i\) are the eigenvalues of the matrix \(R_{m_ n}\), \[ \alpha_ i= v_ i(1-\gamma)+ \gamma v^ 2_ i m^{-1}\sum^ m_{p=1} (v_ i- \mu_ p)^{-1},\quad i=1,2,\;\gamma= m_ n n^{- 1}, \] and \(v_ 1\) and \(v_ 2\) are the maximal and minimal solutions of the equation \[ 1- \gamma+ 2\gamma v m^{-1} \sum^ m_{p=1} (v- \mu_ p)^{-1}= \gamma v^ 2 m^{-1} \sum^ n_{p=1} (v- \mu_ p)^{- 2}. \] {}.

MSC:

62H99	Multivariate analysis
62G30	Order statistics; empirical distribution functions
62E20	Asymptotic distribution theory in statistics

Keywords:

maximum likelihood estimators; empirical coveriance matrices; eigenvalues; maximal and minimal solutions