[Paper Review] An asymptotic distribution theory for Eulerian recurrences with applications

Hsien‐Kuei Hwang, Hua‐Huai Chern+1 more|arXiv (Cornell University)|Jul 4, 2018

Advanced Combinatorial Mathematics204 references4 citations

TL;DR

This paper develops a comprehensive asymptotic distribution theory for Eulerian-type linear recurrences of the form $ P_n(v) = (\alpha(v)n + \gamma(v))P_{n-1}(v) + \beta(v)(1-v)P_{n-1}'(v) $, using the method of moments and analytic combinatorics. It establishes a wide array of limit laws—including normal, beta, Rayleigh, Poisson, and Mittag-Leffler—across over 500 concrete examples, revealing surprising richness in a simple recurrence framework and providing convergence rates via Berry-Esseen bounds.

ABSTRACT

We study linear recurrences of Eulerian type of the form \[ P_n(v) = (α(v)n+γ(v))P_{n-1}(v) +β(v)(1-v)P_{n-1}'(v)\qquad(n\ge1), \] with $P_0(v)$ given, where $α(v), β(v)$ and $γ(v)$ are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of $P_n(v)$ for large $n$ using the method of moments and analytic combinatorial tools under varying $α(v), β(v)$ and $γ(v)$, and apply our results to more than two hundred of concrete examples when $β(v) e0$ and more than three hundred when $β(v)=0$ that we gathered from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.

Motivation & Objective

• To develop a general asymptotic distribution theory for Eulerian recurrences with polynomial coefficients in $\alpha(v)$, $\beta(v)$, and $\gamma(v)$.
• To characterize the limiting distributions of coefficients in $P_n(v)$ for large $n$ across diverse parameter regimes.
• To unify and extend known limit laws (e.g., normal, beta, Poisson) arising from permutation statistics and Eulerian polynomials.
• To provide convergence rates, including Berry-Esseen bounds, for the central limit theorems derived.
• To systematically catalog and analyze over 500 concrete examples from the literature and OEIS, identifying their limiting distributions.

Proposed method

• Employing the method of moments to analyze the asymptotic behavior of central moments of coefficient distributions in $P_n(v)$.
• Using partial differential equations (PDEs) derived from the recurrence to model the generating function dynamics.
• Applying singularity analysis and quasi-powers approximation to extract asymptotic distributions from generating functions.
• Leveraging complex-analytic tools and the quasi-powers theorem to establish central limit theorems under varying $\alpha(v)$, $\beta(v)$, and $\gamma(v)$.
• Introducing a parametrization framework $\mathscr{E}_k\langle\!\langle \alpha(v), \beta(v); \gamma(v) \rangle\!\rangle$ to classify and analyze recurrence families.
• Extending results to non-homogeneous recurrences, recurrences involving $P_{n-2}(v)$, and systems of recurrences to broaden applicability.

Experimental results

Research questions

• RQ1 What limit laws emerge from Eulerian recurrences of the form $ P_n(v) = (\alpha(v)n + \gamma(v))P_{n-1}(v) + \beta(v)(1-v)P_{n-1}'(v) $ as $n \to \infty$?
• RQ2 How do the parameters $\alpha(v)$, $\beta(v)$, and $\gamma(v)$—especially when polynomials of low degree—affect the type and parameters of the limiting distribution?
• RQ3 What convergence rates (e.g., Berry-Esseen bounds) can be established for the central limit theorems derived from such recurrences?
• RQ4 Can non-normal limit laws such as beta, Rayleigh, or Mittag-Leffler distributions arise in this framework, and under what conditions on $\alpha(v)$ and $\beta(v)$?
• RQ5 How can this theory be systematically applied to classify and predict the limiting behavior of known sequences from OEIS and the literature?

Key findings

• For $ (\alpha(v), \beta(v)) = (qv, qv) $, the distribution of coefficients converges to $ \mathscr{N}\bigl(\frac{1}{2}n, \frac{1}{12}n\bigr) $, with convergence rate $ O(n^{-1/2}) $.
• When $ (\alpha(v), \beta(v)) = (qv, v) $, the limit is $ \mathscr{N}\bigl(\frac{q}{q+1}n, \frac{q^2}{(q+1)^2(q+2)}n\bigr) $, with explicit variance and $ O(n^{-1/2}) $ convergence.
• For $ (\alpha(v), \beta(v)) = (v^2, v(1+v)) $, the limit is $ \mathscr{N}\bigl(\frac{2}{3}n, \frac{8}{45}n\bigr) $, demonstrating normality even with quadratic coefficients.
• When $ \beta(v) = 0 $, non-normal laws emerge: for $ (\alpha(v), \beta(v)) = (p+qv, 0) $, the limit is $ \mathscr{N}\bigl(\frac{q}{p+q+1}n, \frac{q(p+1)(p+q)}{(p+q+1)^2(p+q+2)}n\bigr) $, showing broad applicability.
• For $ \frac{\beta}{\alpha} = -1 $, the limit is a beta distribution, including uniform $ \text{Beta}(1,1) $ and arcsine $ \text{Beta}(\frac{1}{2},\frac{1}{2}) $ laws.
• For $ \frac{\beta}{\alpha} = -\frac{1}{2} $, the limit is Rayleigh or half-normal, with examples like $ (\alpha(v), \beta(v)) = (\frac{1}{2}(1+v^2), \frac{1}{2}(1+v^2)) \Rightarrow \mathscr{N}(\frac{1}{2}n, \frac{5}{12}n) $.