  In this paper we study the distribution tails and the moments of a condition
number which arises in the study of homogeneous systems of linear inequalities.
We consider the case where this system is defined by a Gaussian random matrix
and characterise the exact decay rates of the distribution tails, improve the
existing moment estimates, and prove various limit theorems for large scale
systems. Our results are of complexity theoretic interest, because
interior-point methods and relaxation methods for the solution of systems of
linear inequalities have running times that are bounded in terms of the
logarithm and the square of the condition number respectively.
