  We consider the Student-t and Student-r distributions, which maximise Renyi
entropy under a covariance condition. We show that they have
information-theoretic properties which mirror those of the Gaussian
distributions, which maximise Shannon entropy under the same condition. We
introduce a convolution which preserves the Renyi maximising family, and show
that the Renyi maximisers are the case of equality in a version of the Entropy
Power Inequality. Further, we show that the Renyi maximisers satisfy a version
of the heat equation, motivating the definition of a generalized Fisher
information.
