[Paper Review] A central limit theorem for descents and major indices in fixed conjugacy classes of $S_n$

Gene B. Kim, Sang‐Chul Lee|ArXiv.org|Nov 12, 2018

Advanced Combinatorial Mathematics10 references4 citations

TL;DR

This paper establishes a central limit theorem for the joint distribution of descents and major indices in fixed conjugacy classes of the symmetric group $S_n$. By modifying Curtiss' theorem on moment generating functions and deriving a uniform estimate for the joint generating function, the authors prove that the normalized descent and major index statistics converge to a bivariate normal distribution, with parameters depending on the fixed-point density of the conjugacy class.

ABSTRACT

The distribution of descents in fixed conjugacy classes of $S_n$ has been studied, and it is shown that its moments have interesting properties. Kim and Lee showed, by using Curtiss' theorem and moment generating functions, how to prove a central limit theorem for descents in arbitrary conjugacy classes of $S_n$. In this paper, we prove a modified version of Curtiss' theorem to shift the interval of convergence in a more convenient fashion and use this to show that the joint distribution of descents and major indices is asymptotically bivariate normal.

Motivation & Objective

• To establish a central limit theorem for the joint distribution of descents and major indices in fixed conjugacy classes of $S_n$.
• To extend prior results on asymptotic normality of descents to the bivariate case involving both descents and major indices.
• To develop a modified version of Curtiss' theorem for moment generating functions to handle convergence in distribution with improved convergence intervals.
• To provide a uniform estimate on the moment generating function of the joint distribution, enabling generalization to conjugation-invariant subsets with fixed-point density constraints.

Proposed method

• Derive a closed-form expression for the joint generating function of descents and major indices over a conjugacy class $\mathcal{C}_\lambda$ using combinatorial generating function techniques.
• Introduce a modified version of Curtiss' theorem that allows pointwise convergence of moment generating functions to imply convergence in distribution, with enhanced control over the domain of convergence.
• Apply a change of variables and asymptotic analysis to transform the generating function into a Gaussian integral form, enabling estimation of the moment generating function of the normalized statistics.
• Establish a uniform bound on the moment generating function of the normalized descent and major index vector, with error term $\mathcal{O}(n^{-1/6})$, under the assumption of fixed-point density $\alpha_{1,n} \to \alpha \in [0,1]$.
• Use the uniform estimate to prove convergence in distribution to a bivariate normal law with zero mean and covariance matrix $\Sigma_\alpha$ depending only on the limiting fixed-point density $\alpha$.
• Generalize the result to conjugation-invariant subsets $A_n \subset S_n$ where all elements have the same number of fixed points, under the same limiting density condition.

Experimental results

Research questions

• RQ1 Does the joint distribution of descents and major indices in a fixed conjugacy class of $S_n$ converge to a bivariate normal distribution as $n \to \infty$?
• RQ2 Can a modified version of Curtiss' theorem be used to establish convergence in distribution for multivariate statistics when the moment generating function converges pointwise on a shifted domain?
• RQ3 What is the asymptotic mean and covariance structure of the joint distribution of descents and major indices in conjugacy classes with fixed-point density $\alpha_{1,n} \to \alpha$?
• RQ4 How does the presence of fixed points in permutations affect the limiting distribution of descent and major index statistics?
• RQ5 Can the result be extended to broader classes of conjugation-invariant subsets of $S_n$ beyond conjugacy classes?

Key findings

• The joint distribution of normalized descents and major indices in any fixed conjugacy class of $S_n$ converges in distribution to a bivariate normal law with zero mean and covariance matrix $\Sigma_\alpha$ depending only on the limiting fixed-point density $\alpha$.
• The limiting mean of the normalized descent statistic is $\frac{1 - \alpha^2}{2}n$, and the limiting mean of the normalized major index is $\frac{1 - \alpha^2}{4}n^2$, with scaling factors $n^{1/2}$ and $n^{3/2}$ respectively.
• A modified version of Curtiss' theorem is developed, which allows convergence of moment generating functions on a shifted interval to imply convergence in distribution, enhancing applicability to multivariate asymptotic analysis.
• A uniform estimate on the moment generating function of the joint distribution is established with error $\mathcal{O}(n^{-1/6})$, which is sufficient to prove the central limit theorem.
• The result extends to any conjugation-invariant subset $A_n \subset S_n$ where all elements have the same number of fixed points and the fixed-point density $\alpha_{1,n} \to \alpha \in [0,1]$, yielding the same limiting bivariate normal distribution.
• The authors conjecture that the $1 - \alpha^2$ factor in the asymptotic means may have a deeper combinatorial interpretation, which remains to be explained.