Stein's method and the rank distribution of random matrices over finite fields

Abstract

With ${\mathcal{Q}}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times (n+m)$ matrices over the finite field ${\mathbb{F}}_q$ of size $q\ge 2$, and ${\mathcal{Q}}_q$ the distributional limit of ${\mathcal{Q}}_{q,n}$ as $n\to \infty$, we apply Stein's method to prove the total variation bound $\frac{1}{8q^{n+m+1}}\le \Vert {\mathcal{Q}}_{q,n}-{\mathcal{Q}}_q{\Vert }_{\mathrm{TV}}\le \frac{3}{q^{n+m+1}}$. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.

Published at http://dx.doi.org/10.1214/13-AOP889 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Discrete Stein characterizations and discrete information distances

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