Law of large numbers and central limit theorem for Jordan normal form of large triangular matrices over a finite field

We prove that for a typical stricly uppertriangular matrix of order $n$ over a finite field with $q$ elements the sequence of orders of Jordan blocks, divided by $n$, converges to the geometric progression $\{(q-1)q^{-k},\,k=1, 2,\dots\}$, $n\to\infty$. We also show that the distribution of orders for a finite number of Jordan blocks is asymptotically normal. The corresponding covariance matrix is calculated.