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

Girko, V. L.

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

Keywords:

asymptotics of eigenvalues; empirical covariance matrices; resolvent; normally distributed vector; Stieltjes transform

Citations:

Zbl 0788.62054