STEIN'S METHOD AND THE RANK DISTRIBUTION OF RANDOM MATRICES OVER FINITE FIELDS

With Q(q,n) the distribution of n minus the rank of a matrix chosen uniformly from the collection of all n x (n + m) matrices over the finite field F-q of size q >= 2, and Q(q) the distributional limit of Q(q,n) as n -> infinity, we apply Stein's method to prove the total variation bound 1/8q(n+m+1) <= parallel to Q(q,n) - Qq parallel to TV <= 3/q(n+m+1). In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.