  The transition probability of a Cox-Ingersoll-Ross process can be represented
by a non-central chi-square density. First we prove a new representation for
the central chi-square density based on sums of powers of generalized Gaussian
random variables. Second we prove Marsaglia's polar method extends to this
distribution, providing a simple, exact, robust and efficient
acceptance-rejection method for generalized Gaussian sampling and thus central
chi-square sampling. Third we derive a simple, high-accuracy, robust and
efficient direct inversion method for generalized Gaussian sampling based on
the Beasley-Springer-Moro method. Indeed the accuracy of the approximation to
the inverse cumulative distribution function is to the tenth decimal place. We
then apply our methods to non-central chi-square variance sampling in the
Heston model. We focus on the case when the number of degrees of freedom is
small and the zero boundary is attracting and attainable, typical in foreign
exchange markets. Using the additivity property of the chi-square distribution,
our methods apply in all parameter regimes.
