A rate of convergence result for the largest eigenvalue of complex white Wishart matrices

By Noureddine El Karoui. 2004

85 citations recorded by the source corpus.

It has been recently shown that if X is an n×N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X*X, there exist sequences mn,N and sn,N such that (l1−mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F2. In this paper we show that, under the assumption that n/N→ γ∈(0, ∞), we can find a function M, continuous and nonincreasing, and sequences μn,N and σn,N such that, for all real s0, there exists an integer N(s0, γ) for which, if (n∧N)≥N(s0, γ), we have, with ln,N=(l1−μn,N)/σn,N, ∀ s≥s0 (n∧N)2/3|P(ln,N≤s)−F2(s)|≤M(s0)exp(−s). The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (e.g., statistical) applications.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/764e516256c6f7141026499187df880dc465b565

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Citations in the corpus

Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices (2005, background): “We refer to [11] for theoretical considerations that help understand why this is happening and some simulations in the complex Wishart case.”
Non-Parametric Detection of the Number of Signals: Hypothesis Testing and Random Matrix Theory (2009, background): “While there is no simple explicit expression for the distribution of its largest eigenvalue, in recent years, its exact distribution under a certain asymptotic limit was derived [7], [14], [15] : Theorem 1: Let denote the sample covariance matrix of pure noise vectors distributed .”
Determining the number of components in a factor model from limited noisy data (2008, background): “For complex-valued observations the definitions of μn,p and σn,p giving the same convergence rate are more involved and appear in [14].”
Universality results for largest eigenvalues of some sample covariance matrix ensembles (2007, background): “They have been established in [8] for complex Gaussian samples.”
High Dimensional Statistical Inference and Random Matrices (2006, unknown): “The second-order result (7) is due to El Karoui [38], If μ′np and σ ′ np denote the quantities in (8) with n and p replaced by n + 1/2 and p + 1/2, then the centering μnp is a weighted combination of μ′n−1,p and μ′n,p−1 and the scaling σnp a similar combination of σ ′ n−1,p and σ ′ n,p−1.”
Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices (2012, methodology): “The alignment in [10], equation (5) and A.”