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

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