## Limiting spectral distribution for a class of random matrices.(English)Zbl 0614.62060

Let $$X=\{X_{ij}$$, $$i,j=1,2,...\}$$ be an infinite dimensional random matrix of i.i.d. entries and E $$X^ 2_{11}<\infty$$. Let $$T_ p$$ be a $$p\times p$$ nonnegative definite random matrix independent of X, for $$p=1,2,...$$. Suppose $$p^{-1}tr T^ k_ p\to H_ k$$ a.s. as $$p\to \infty$$ for $$k=1,2,...$$, and $$\sum H_{2k}^{-(2k)^{-1}}<\infty$$. Then the spectral distribution of $$A_ p=n^{-1}X_ pX_ p'T_ p$$, where $$X_ p=\{X_{ij}$$, $$i=1,...,p$$, $$j=1,...,n\}$$ tends to a nonrandom limit distribution as $$p\to \infty$$, $$n\to \infty$$, but $$p/n\to y>0.$$
The two-stage truncation method was used in proving the main result. The notion of M-graphs was introduced and some properties were derived.
Reviewer: M.Huškova

### MSC:

 62H10 Multivariate distribution of statistics 15B52 Random matrices (algebraic aspects)
Full Text:

### References:

  Bai, Z.D.; Yin, Y.Q.; Krishnaiah, P.R., On limiting spectral distribution of product of two random matrices when the underlying distribution is isotropic, J. multivariate anal., 19, 189-200, (1986) · Zbl 0657.62058  Dudley, R.M., Convergence of Baire measures, Studia math., 27, 251-268, (1966) · Zbl 0147.31301  Fan, K., Maximum properties and inequalities for the eigenvalues of completely continuous operators, (), 760-766 · Zbl 0044.11502  Grenander, V.; Silverstein, J., Spectral analysis of networks with random topologies, SIAM J. appl. math., 32, 499-519, (1977) · Zbl 0355.94043  Hoeffding, W., Probability inequalities for sums of bounded random variables, J. amer. statist. assoc., (1963) · Zbl 0127.10602  Von Neumann, J., Some matrix-inequalities and metrization of matric space, Tomsk univ. rev., 1, 286-300, (1937) · Zbl 0017.09802  Wachter, K.W., The strong limits of random matrix spectra for sample matrices of independent elements, Ann. probab., 6, 1-18, (1978) · Zbl 0374.60039  Yin, Y.Q.; Krishnaiah, P.R., A limit theorem for the eigenvalues of product of two random matrices, J. multivariate anal., 13, 489-507, (1984) · Zbl 0553.62018  Yin, Y.Q.; Krishnaiah, P.R., Limit theorems for the eigenvalues of the sample covariance matrix when the underlying distribution is isotropic, (), in press · Zbl 0658.62025  Bai, Z.D.; Liang, W.Q., (), Tech. Rep. 1985
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.