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Indian Journal of Pure and Applied Mathematics

, Volume 48, Issue 4, pp 575–607 | Cite as

Matrix polynomial generalizations of the sample variance-covariance matrix when pn−1y ∈ (0, ∞)

  • Monika BhattacharjeeEmail author
  • Arup Bose
Article
  • 67 Downloads

Abstract

Let {Z u = ((εu, i, j))p×n} be random matrices where {εu, i, j} are independently distributed. Suppose {A i }, {B i } are non-random matrices of order p × p and n × n respectively. Consider all p × p random matrix polynomials \(P = \prod\nolimits_{i = 1}^{k_l } {\left( {n^{ - 1} A_{t_i } Z_{j_i } B_{s_i } Z_{j_i }^* } \right)A_{t_{k_l + 1} } }\). We show that under appropriate conditions on the above matrices, the elements of the non-commutative *-probability space Span {P} with state p−1ETr converge. As a by-product, we also show that the limiting spectral distribution of any self-adjoint polynomial in Span{P} exists almost surely.

Key words

Independent matrix moment method Stieltjes transformation limiting spectral distribution semi-circle law non-crossing partition non-commutative probability space *-algebra free cumulants 

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Copyright information

© The Indian National Science Academy 2017

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteKolkataIndia

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