Eigenvalue statistics for the sum of two complex Wishart matrices

Abstract

The sum of independent Wishart matrices, taken from distributions with unequal covariance matrices, plays a crucial role in multivariate statistics, and has applications in the fields of quantitative finance and telecommunication. However, analytical results concerning the corresponding eigenvalue statistics have remained unavailable, even for the sum of two Wishart matrices. This can be attributed to the complicated and rotationally noninvariant nature of the matrix distribution that makes extracting the information about eigenvalues a nontrivial task. Using a generalization of the Harish-Chandra-Itzykson-Zuber integral, we find exact solution to this problem for the complex Wishart case when one of the covariance matrices is proportional to the identity matrix, while the other is arbitrary. We derive exact and compact expressions for the joint probability density and marginal density of eigenvalues. The analytical results are compared with numerical simulations and we find perfect agreement.

Published version

References in corpus (24)

• Noise Dressing of Financial Correlation Matrices
• A Random Matrix Approach to Cross-Correlations in Financial Data
• Random matrix theory and spectral sum rules for the Dirac operator in QCD
• Matrix Models for Beta Ensembles
• The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way
• Induced measures in the space of mixed quantum states
• Random Matrix Theory and Chiral Symmetry in QCD
• Biorthogonal ensembles
• MIMO Interference Alignment Over Correlated Channels with Imperfect CSI
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Testing statistical bounds on entanglement using quantum chaos
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation
• Quantum conductance problems and the Jacobi ensemble
• Capacity and Character Expansions: Moment generating function and other exact results for MIMO correlated channels
• Universal spectral correlations of the Dirac operator at finite temperature
• On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices
• Probability densities and distributions for spiked and general variance Wishart $\beta$-ensembles
• Random Matrix Model for Nakagami-Hoyt Fading
• Universal Correlations and Power-Law Tails in Financial Covariance Matrices
• Distribution of the Smallest Eigenvalue in the Correlated Wishart Model
• Supersymmetry approach to Wishart correlation matrices: Exact results
• Credit Risk and the Instability of the Financial System: an Ensemble Approach
• Non-Hermitian extensions of Wishart random matrix ensembles

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• On the Exact and Approximate Eigenvalue Distribution for Sum of Wishart Matrices
• Spectral statistics for the difference of two Wishart matrices
• A variant of Horn's problem and derivative principle
• Duistermaat-Heckman measure and the mixture of quantum states
• Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type
• Universal Broadening of Zero Modes: A General Framework and Identification