Polynomial generalizations of the sample variance-covariance matrix when p n-1 → 0
Article Type
Research Article
Publication Title
Random Matrices: Theory and Application
Abstract
Let {Zu = ((ϵu,i,j))p×n} be random matrices, where {ϵu,i,j} are independently distributed. Suppose {Ai}, {Bi} are non-random matrices of order p × p and n × n, respectively. Suppose p →∞, n = n(p) →∞ and p/n → 0. Consider all p × p random matrix polynomials constructed from the above matrices of the form =- i=1kln-1A ti ZjiBsiZji-A tkl+1 and the corresponding centering polynomials = (- i=1kln-1Tr(B si))- i=1kl+1A ti. We show that under appropriate conditions on the above matrices, the variables in the non-commutative-probability space p = Span{(n/p)1/2(-)} with state p-1ETr converge. We also show that the limiting spectral distribution of (n/p)1/2(-) exists almost surely whenever and are self-adjoint. The limit can be expressed in terms of, semi-circular, circular and other families and, limits of n-1Tr(B i), n-1Tr(B iBj) and non-commutative limit of {Ai: i ≥ 1,φp = p-1Tr}. Our results fully generalize the results already known for np-1(n-1A 11/2ZB 1Z-A 11/2-n-1Tr(B 1)A1).
DOI
10.1142/S2010326316500143
Publication Date
10-1-2016
Recommended Citation
Bhattacharjee, Monika and Bose, Arup, "Polynomial generalizations of the sample variance-covariance matrix when p n-1 → 0" (2016). Journal Articles. 4380.
https://digitalcommons.isical.ac.in/journal-articles/4380