Deterministic equivalents for certain functionals of large random matrices

Abstract

Consider an N × n random matrix Yn = (Yijn) where the entries are given by Yijn = σij(n)/√n Xijn being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic N × n matrix A n whose columns and rows are uniformly bounded in the Euclidean norm. Let σn = Yn + An. We prove in this article that there exists a deterministic NxN matrix-valued function T n(z) analytic in ℂ - ℝ+ such that, almost surely, lim n→+∞,N/n→c (1/N Trace (σnσ nT - zIN)-1 - 1/N Trace T n(z)) = 0. Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform, of the distribution of the eigenvalues of σn σnT. For each n, the entries of matrix Tn (z) are defined as the unique solutions of a certain system of nonlinear functional equations. It is also proved that 1/N Trace T n(z) is the Stieltjes transform, of a probability measure πn(dλ), and that for every bounded continuous function f, the following convergence holds almost surely 1/N σk=1N f(λk) - ∫0∞ f(λ)πn(dλ)n → 0, n → ∞ where the (λk)1≤k≤N are the eigenvalues of σσnT. This work is motivated by the context of performance evaluation of multiple inputs/multiple output (MIMO) wireless digital communication channels. As an application, we derive a deterministic equivalent to the mutual information: Cn(σ2) = 1/N E log det (1N + σnσnT/σ2), where σ2, is a known parameter. © Institute of Mathematical Statistics, 2007.