Abstract:
LetWbe an n×n complex non-centralWishart matrix with m (≥ n) degrees of
freedom and a rank-1 mean. In this paper, we consider three problems related
to the eigenvalues of W. To be specific, we derive a new expression for the
cumulative distribution function (c.d.f.) of the minimum eigenvalue (λmin) of
W. The c.d.f. is expressed as the determinant of a square matrix, the size of
which depends only on the difference m−n. This further facilitates the analysis
of the microscopic limit of the minimum eigenvalue. The microscopic limit
takes the form of the determinant of a square matrix with its entries expressed
in terms of the modified Bessel functions of the first kind. We also develop a
moment generating function based approach to derive the probability density
function of the random variable tr(W)/λmin, where tr(·) denotes the trace of
a square matrix. Moreover, we establish that, as m, n → ∞ with m − n fixed,
tr(W)/λmin scales like n3. Finally, we find the average of the reciprocal of the
characteristic polynomial det[zIn +W], | arg z| < π, where In and det[·] denote
the identity matrix of size n and the determinant, respectively.
Citation:
Dharmawansa, P. (2016). Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean. Journal of Multivariate Analysis, 149, 30–53. https://doi.org/10.1016/j.jmva.2016.03.003
