The law of large numbers and the central limit theorem for the Jordan normal form of large triangular matrices over a finite field
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.
versions of law of large numbers and central limit theorem
scientific article; zbMATH DE number 1408550