[Paper Review] A central limit theorem for the number of descents and some urn models

Olivier Garet|arXiv (Cornell University)|Jun 7, 2019

Stochastic processes and statistical mechanics6 references4 citations

TL;DR

This paper establishes a general Central Limit Theorem (CLT) for a class of Markov chains with specific conditional moment structures, applicable to combinatorial and urn models such as the number of descents in random permutations and generalized Pólya urns. The key result shows that the normalized sum $ S_n $ converges to a normal distribution with variance $ S = \frac{D}{2\alpha_1 + 1} $, where $ D = D_2 - \ell(\ell + \alpha_2) $, $ \ell = \frac{D_1}{\alpha_1 + 1} $, under conditions $ \alpha_1 > -\frac{1}{2} $ and $ D > 0 $. The theorem unifies and extends CLTs for descent counts and various urn dynamics.

ABSTRACT

The purpose of this work is to establish a central limit theorem that can be applied to a particular form of Markov chains, including the number of descents in a random permutation of $\mathfrak{S}_n$, two-type generalized P{ó}lya urns, and some other urn models.

Motivation & Objective

• To develop a unified Central Limit Theorem applicable to a broad class of Markov chains arising in combinatorial and stochastic processes.
• To establish conditions under which the normalized sum of dependent random variables converges to a normal distribution, even when the increments are not i.i.d.
• To provide a theoretical framework that generalizes known CLTs for the number of descents in random permutations and two-type Pólya urns.
• To analyze the asymptotic behavior of urn models with deterministic replacement rules and structural constraints, such as black ball dominance.
• To derive explicit expressions for the limiting variance in terms of model parameters, enabling quantitative predictions in applications.

Proposed method

• The method relies on conditional expectation and the method of moments, analyzing the first three conditional moments of the increment process.
• The model assumes that the conditional $ k $-th moment of $ a_{n+1} $, given past values, follows a linear form in $ S_n $, with coefficients converging to constants $ D_k $ and $ \alpha_k $ as $ n \to \infty $.
• The key technical tool is a recurrence analysis using analytic lemmas involving Pochhammer-like products and asymptotic expansions of sequences.
• The proof leverages telescoping sums and Stirling’s approximation to solve the recurrence for the first moment, leading to the deterministic limit $ \ell = \frac{D_1}{\alpha_1 + 1} $.
• The variance of the limiting normal distribution is derived as $ S = \frac{D}{2\alpha_1 + 1} $, where $ D = D_2 - \ell(\ell + \alpha_2) $, under regularity conditions.
• The framework is applied to the insertion process of random permutations and to a novel urn model with circular arrangement and three-ball replacement rules, both satisfying the moment conditions.

Experimental results

Research questions

• RQ1 Under what general conditions does the sum of a Markovian sequence of bounded random variables converge to a normal distribution?
• RQ2 Can the classical CLT for the number of descents in a random permutation be derived from a more general stochastic framework?
• RQ3 How can the limiting variance in generalized Pólya urn models be explicitly computed from the replacement rule and initial conditions?
• RQ4 What structural constraints ensure the long-term dominance of one color in a circular urn model with three-ball replacement?
• RQ5 Does the CLT hold even when the process is not i.i.d. or martingale, but satisfies a specific conditional moment structure?

Key findings

• The normalized number of descents in a uniformly random permutation of $ \mathfrak{S}_n $ satisfies $ \frac{|\mathrm{Desc}(\sigma_n)| - n/2}{\sqrt{n}} \xrightarrow{d} \mathcal{N}\left(0, \frac{1}{12}\right) $, which is recovered as a special case.
• For the urn model with $ a=2, b=3 $, the limiting distribution is $ \frac{S_n - \frac{4}{5}n}{\sqrt{n}} \xrightarrow{d} \mathcal{N}\left(0, \frac{7}{50}\right) $, with $ S_n $ being the number of white balls.
• The limiting variance is given by $ S = x(5 - 9x)\frac{C - 3}{C + 3} $, where $ x = \frac{a}{a+b} $, $ C = a + b \geq 4 $, and $ x \in (0, \frac{1}{2}) $, ensuring $ D > 0 $.
• The condition $ \alpha_1 > -\frac{1}{2} $ ensures the stability of the mean behavior, and $ D > 0 $ guarantees non-degenerate limiting variance.
• The method applies to models with deterministic replacement rules and structural constraints (e.g., black ball dominance), even when the process is not a martingale.
• The convergence in distribution to normality is established via the method of moments, and almost sure convergence remains an open question.