Balanced weighted Motzkin paths: Pearson structure and saddlepoint asymptotics

TL;DR

This paper studies weighted Motzkin paths with height-dependent step multiplicities, deriving explicit formulas, limit distributions, large deviations, and saddlepoint approximations, linking Pearson geometry with advanced asymptotic methods.

Contribution

It provides closed-form expressions, limit theorems, and saddlepoint approximations for balanced weighted Motzkin paths, extending the understanding of their asymptotic behaviour.

Findings

Gaussian central window for terminal-height distribution

Explicit limit cumulant generating function identified

Uniform saddlepoint approximation with order n^{-1} accuracy

Abstract

We analyse weighted Motzkin paths with step multiplicities that vary linearly with height. In the balanced case the associated exponential generating function satisfies a Pearson-type PDE, and solving by characteristics yields closed expressions in all drift regimes. These formulas reveal a moving algebraic singularity that governs both local and global behaviour. Locally this gives a Gaussian central window for the terminal-height distribution, while globally we identify an explicit limit cumulant generating function and prove an $n$-speed large-deviation principle. For finite $n$, Daniels' lattice saddlepoint approximation provides a single formula that is accurate across the full range of $k$; in all quadratic regimes it achieves a uniform interior relative error of order $n^{-1}$. The results link Pearson geometry with uniform saddlepoint methods and extend naturally to other weighted path models and tridiagonal recurrences.