The law of large numbers and the central limit theorem for the Jordan normal form of large triangular matrices over a finite field
type: Item
instance of: scholarly article
title: The law of large numbers and the central limit theorem for the Jordan normal form of large triangular matrices over a finite field
zbMATH Open document ID: 0952.60020
author: Alexei Borodin
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-03-06T00:00:00Z
review text: For \(n \in {\mathbb N}\), consider a set \(\Omega_n = g_n({\mathbb F}_q)\) of \(n\times n\) upper triangular matrices \(X\) over a finite \(q\)-element field \({\mathbb F}_q\) with uniform probability \(P^{(n)}(X) = q^{-n(n-1)/2}\). A partition \(\lambda = \{\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_r \}\) of \(n\) is the partition type of \(X\) if \(\lambda_k\) are the orders of the Jordan blocks of the normal form of \(X\). Viewing \(\lambda_k\) as random variables \(\lambda_k^{(n)}\) on \((\Omega_n,P^{(n)})\), the paper gives a result in the spirit of LLN: for any \(\varepsilon > 0\) there exist constants \(c=c(\varepsilon) > 0\) and \(\theta = \theta(\varepsilon) \in (0,1)\) such that \[ P^{(n)}\Bigl\{\sup_k|\lambda_k^{(n)}n^{-1} - (q^{1-k} - q^{-k})|\geq \varepsilon \Bigr\} \leq c\theta^n,\quad n \in {\mathbb N}. \] This entails also a version of CLT. The asymptotics in \(n\) is analyzed with the help of Young diagrams corresponding to \(\lambda\) when \(X \in \Omega_n\) is embedded into \(\Omega_{n+1}\). As a corollary an inequality involving the dimension of irreducible representation of symmetric group \(S_n\) determined by \(\lambda\) is obtained.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1408550
zbMATH Keywords: matrices
zbMATH Keywords: Young diagrams
zbMATH Keywords: versions of law of large numbers and central limit theorem
reviewed by: Alexander V. Bulinski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1408550