  We introduce a one-parameter deformation of the Wishart-Laguerre or chiral
ensembles of positive definite random matrices with Dyson index beta=1,2 and 4.
Our generalised model has a fat-tailed distribution while preserving the
invariance under orthogonal, unitary or symplectic transformations. The
spectral properties are derived analytically for finite matrix size NxM for all
three beta, in terms of the orthogonal polynomials of the standard
Wishart-Laguerre ensembles. For large-N in a certain double scaling limit we
obtain a generalised Marcenko-Pastur distribution on the macroscopic scale, and
a generalised Bessel-law at the hard edge which is shown to be universal. Both
macroscopic and microscopic correlations exhibit power-law tails, where the
microscopic limit depends on beta and the difference M-N. In the limit where
our parameter governing the power-law goes to infinity we recover the
correlations of the Wishart-Laguerre ensembles. To illustrate these findings
the generalised Marcenko-Pastur distribution is shown to be in very good
agreement with empirical data from financial covariance matrices.
