Some stochastic processes in a random permutation (Q1105267)

For a random permutation \(\sigma\) of \(\{\) 1,2,...,N\(\}\), an increasing success at i is the event that \(\sigma (i)=j\), \(\sigma (k)=j-1\) for some \(k<i\) and \(j\geq 2\), and a decreasing success the event that \(\sigma (i)=h\), \(\sigma (k)=h+1\) for some \(k<i\) and \(h\leq N-1\). Three processes of events which have the same distribution are considered. The first of them is defined in terms of rises and descents of \(\sigma\), the second one on a random circular permutation of \(\{1,2,...,N+1\}\) and the third one on a chess board model. Next, the distributions of some statistics of these processes are given and their asymptotic behaviour as \(N\to \infty\) is studied.