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May 2015 Stein’s method and the rank distribution of random matrices over finite fields
Jason Fulman, Larry Goldstein
Ann. Probab. 43(3): 1274-1314 (May 2015). DOI: 10.1214/13-AOP889

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 $q\ge2$, and $\mathcal{Q}_{q}$ the distributional limit of $\mathcal{Q}_{q,n}$ as $n\rightarrow\infty$, we apply Stein’s method to prove the total variation bound

\[\frac{1}{8q^{n+m+1}}\leq\|\mathcal{Q}_{q,n}-\mathcal{Q}_{q}\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}.\] In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.

Citation

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Jason Fulman. Larry Goldstein. "Stein’s method and the rank distribution of random matrices over finite fields." Ann. Probab. 43 (3) 1274 - 1314, May 2015. https://doi.org/10.1214/13-AOP889

Information

Published: May 2015
First available in Project Euclid: 5 May 2015

zbMATH: 06455733
MathSciNet: MR3342663
Digital Object Identifier: 10.1214/13-AOP889

Subjects:
Primary: 60B20, 60C05, 60F05

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 3 • May 2015
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