The convergence of random quadratic forms with application to Gaussian Orthogonal and Wishart Ensembles

Abstract

We consider the convergence of the quadratic form in independent, centred random variables and a real symmetric matrix V, with a non-vanishing diagonal. Its central limit theorem requires a Lindeberg-type condition on V 's eigenvalues. 

We derive sufficient conditions in terms of V 's eigenvalues and eigenvectors for convergence to a weighted sum of chi-squared random variables, with weights given by the eigenvalues. One of the conditions is that V must have an orthonormal eigenvector matrix that is a rotationally invariant random matrix in distribution. This is, for example, the case if V follows the Gaussian Orthogonal Ensemble or the white Wishart ensemble.

The Wishart ensemble idealises the covariance matrix used in finance to calculate portfolio tracking error variance. Consequently, the quadratic form's asymptotic distribution function gives insight into the distribution of tracking error variance for large portfolios.

Keywords: Random quadratic form, Convergence theorems, Central limit theorem, Weighted sum of chi-squared random variables, Gaussian Orthogonal Ensemble, White Wishart ensemble, Rotationally invariant random matrices