## The asymptotics of the eigenvalues of empirical covariance matrices. II.(English. Russian original)Zbl 0834.62017

Theory Probab. Math. Stat. 46, 33-37 (1993); translation from Teor. Jmovirn. Mat. Stat. 46, 36-42 (1992).
Summary: [For part I see Theory Probab. Math. Stat. 44, 37-44 (1992); translation from Teor. Veroyatn. Mat. Stat., Kiew 44, 40-48 (1991; Zbl 0788.62054).]
Suppose $$\varlimsup_{n\to \infty} m_n n^{-1} <1$$ where $$n$$ is the number of independent observations $$x_1, \dots, x_n$$ of an $$m_n$$-dimensional random vector $$\xi$$ that is distributed according to a normal law $$N(a, R_{m_n})$$, $$\mu_{m_n} \leq \cdots\leq \mu_1<c <\infty$$ (where $$\mu_i$$ are the eigenvalues of the matrix $$R_{m_n}$$), and $$\Theta_{n, \varepsilon}$$ is an arbitrary measurable solution of the equation $\mu_1 (\widehat {R})= \theta(1- \gamma)+ \gamma \theta^2 \text{ Re} [f(\theta+ i\varepsilon) (\theta+ i\varepsilon )^{-1} -1- \gamma](\theta+ i\varepsilon )^{-1},$ where $$f(\theta+ i\varepsilon)$$ is an analytic function that satisfies the equation $1- \gamma+ \gamma fm^{-1} \text{ tr} [I(f+ i\varepsilon \text{ sgn Re } f)- \widehat {R} ]^{-1}= (\theta+ i\varepsilon )^{- 1} f,$
$\widehat {R}= n^{-1} \sum_{k=1}^n (x_k- \widehat {x}) (x_k- \widehat {x})', \qquad \widehat {x}= n^{-1} \sum_{k=1}^n x_k, \qquad \gamma= mn^{-1}.$ It is proved that $\lim_{\varepsilon\to 0}\underset{n\to\infty}{\text{p- lim}}[\theta_{n, \varepsilon}- \mu_1 ]=0.$

### MSC:

 62E20 Asymptotic distribution theory in statistics 65C99 Probabilistic methods, stochastic differential equations 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A18 Eigenvalues, singular values, and eigenvectors 62H99 Multivariate analysis

Zbl 0788.62054