Fundamental results of important past mathematicians had applied the strategy of adopting the modified Curtiss’ theorem, and algebraic combinatorial computations had been used to calculate sampling probabilities through numerical simulations of expected joint distribution of random variables in inferential statistics. Different methods of calculation of limits and bounds for estimating functional operators were discovered by Pitman, David and Barton, Tanny, Fulman and many others during the observance of Stein’s discrepancy phenomena reflected among measure sets.
The two main results of this paper concern the convergence in distribution of random variables to the pointwise convergence of their moment generating functions on an open set where a partition of the sampling dimension \(n\) with \(n_i\) parts of size \(i\) relates the density of fixed points, providing the uniform estimate of the discrepancy between a continuous probability distribution and a discrete set of states selected from its domain which authors split in two regions, roughly speaking the small and large range points or regions. These results generalize to a broader class of sequences \((\mathcal{C}_n)\) such that it is a conjugacy-invariant subset of \(S_{n}\) and every element of \(\mathcal{C}_n\) has the same number of fixed points. Finally, various lemmata are proved incorporating the exponentially decaying terms into the deranged and repressed error term completing the proofs. Mean and variance of peaks in the symmetric group are computed to establish the asymptotic normality in \(S_{n}\).
Editorial supplement: The authors’ abstract reads, “The number of peaks of a random permutation is known to be asymptotically normal. We give a new proof of this and prove a central limit theorem for the distribution of peaks in a fixed conjugacy class of the symmetric group. Our technique is to apply analytic combinatorics to study a complicated but exact generating function for peaks in a given conjugacy class.”