The limiting spectral distribution of the noncentral unified matrix model

Abstract We investigate the limiting spectral distribution of a noncentral unified matrix model defined by (X) = ( (XP₁+A) (XP₁+A) '/n₁) (XP₂X'/n₂) ^-1, where X= (X₈₉) ₏ ₍ is a random matrix with independent and identically distributed real entries having zero mean and finite second moment. A is a p n nonrandom matrix. The matrices P₁ and P₂ are projection matrices satisfying rank (P₁) =n₁, rank (P₂) =n₂, and P₁P₂=0. When P₁ and P₂ are random, they are assumed to be independent of X. When p/n₁ c₁ (0, ) and p/n₂ c₂ (0, 1), we establish the almost sure convergence of the empirical spectral distribution of to a deterministic limiting distribution. Furthermore, we show that this limiting distribution coincides with that of the noncentral F-matrix, thus revealing a deep connection between the proposed model and classical multivariate analysis.