Asymptotic Normality of the Generalized Eulerian Numbers

Let \(A_m(n, k)\) denote the number of permutations of \(\{1, \ldots, n\}\) with exactly \(k\) rises of size at least \(m\). We show that, for each positive integer \(m\), the \(A_m(n, k)\) are asymptotically normal.