The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 17, Number 3 (2007), 875-930.
Deterministic equivalents for certain functionals of large random matrices
Consider an N×n random matrix Yn=(Ynij) where the entries are given by , the Xnij being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic N×n matrix An 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 N×N matrix-valued function Tn(z) analytic in ℂ−ℝ+ such that, almost surely,
Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of ΣnΣnT. 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 is the Stieltjes transform of a probability measure πn(dλ), and that for every bounded continuous function f, the following convergence holds almost surely
where the (λk)1≤k≤N are the eigenvalues of ΣnΣnT. 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:
where σ2 is a known parameter.
Ann. Appl. Probab., Volume 17, Number 3 (2007), 875-930.
First available in Project Euclid: 22 May 2007
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems
Hachem, Walid; Loubaton, Philippe; Najim, Jamal. Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 (2007), no. 3, 875--930. doi:10.1214/105051606000000925. https://projecteuclid.org/euclid.aoap/1179839177