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Multifold Convolutions, Generating Functions and 1d Random Walks

Timothy Li, Shannon Starr

TL;DR

This work analyzes multifold convolutions of combinatorial sequences and their probabilistic implications for Dyck paths and simple random walk bridges. By combining generating-function techniques, the circle method, and large-deviation theory, it derives explicit asymptotics and rate functions for returns to the origin, including for Catalan, central-binomial, and higher-fold convolutions like $B_n^2$ and $B_n^3$. The authors introduce implicit rate-function characterizations (via elliptic integrals for 2SRWB) and provide a streamlined circle-method framework that yields concrete asymptotics for $k$-fold convolutions with $k$ growing linearly with $n$. The results connect combinatorial convolution structures with probabilistic limit laws and large-deviation principles, offering explicit formulas and verification through several ensembles and generating-function analyses. This has potential implications for regenerative-process models and the precise asymptotics of return-time distributions in constrained random-walk settings.

Abstract

We consider multifold convolutions of a combinatorial sequence $(a_n)_{n=0}^{\infty}$: namely, for each $k \in \N$ the $k$-fold convolution is $\mathcal{M}^{(k)}_n(\boldsymbol{a}) = \sum_{j_1+\dots+j_k=n} a_{j_1} \cdots a_{j_k}$. Let $C_n$ be the Catalan numbers, and let $B_n$ be the central binomial coefficients. Then for random Dyck paths or simple random walk bridges, the multifold convolutions give moments of returns to the origin, using the stars-and-bars problem. There are well-known explicit formulas for the multifold convolutions of $C_n$ and $B_n$. But even for combinatorial sequences $B_n^2$ and $B_n^3$, one may determine asymptotics of multifold convolutions for large $n$. We also discuss large deviations: In a second part of the paper we consider an elementary version of the circle method for calculating asymptotics using complex analysis.

Multifold Convolutions, Generating Functions and 1d Random Walks

TL;DR

This work analyzes multifold convolutions of combinatorial sequences and their probabilistic implications for Dyck paths and simple random walk bridges. By combining generating-function techniques, the circle method, and large-deviation theory, it derives explicit asymptotics and rate functions for returns to the origin, including for Catalan, central-binomial, and higher-fold convolutions like and . The authors introduce implicit rate-function characterizations (via elliptic integrals for 2SRWB) and provide a streamlined circle-method framework that yields concrete asymptotics for -fold convolutions with growing linearly with . The results connect combinatorial convolution structures with probabilistic limit laws and large-deviation principles, offering explicit formulas and verification through several ensembles and generating-function analyses. This has potential implications for regenerative-process models and the precise asymptotics of return-time distributions in constrained random-walk settings.

Abstract

We consider multifold convolutions of a combinatorial sequence : namely, for each the -fold convolution is . Let be the Catalan numbers, and let be the central binomial coefficients. Then for random Dyck paths or simple random walk bridges, the multifold convolutions give moments of returns to the origin, using the stars-and-bars problem. There are well-known explicit formulas for the multifold convolutions of and . But even for combinatorial sequences and , one may determine asymptotics of multifold convolutions for large . We also discuss large deviations: In a second part of the paper we consider an elementary version of the circle method for calculating asymptotics using complex analysis.

Paper Structure

This paper contains 10 sections, 14 theorems, 140 equations, 2 figures.

Table of Contents

  1. Introduction and first results
  2. A weak LDP for $\mathfrak{Z}_n$ relative to Dyck and SRWB paths
  3. Some observations about the LDP
  4. Implicit formulation of the rate function for the 2SRWB
  5. Proofs of the "first results"
  6. Proof of Theorem \ref{['thm:rate']} using calculus formulas
  7. Proof of calculus formula for Theorem \ref{['thm:rate']}
  8. A simplified circle method
  9. A general derivation of the explicit rate function
  10. Proof of Proposition \ref{['prop:2SRWBrate']} using the generating function

Key Result

Lemma 1.1

For each $k \in \{0,1,\dots\}$ and $n \in \mathbb{N}$,

Figures (2)

  • Figure 1: We plot both large deviation rate functions together: red is for the 2SRWB and green is for the Dyck path. Note that $\ln(2) \approx 0.693147$ and $2\ln(2) \approx 1.38629$.
  • Figure 2: We plot all three large deviation rate functions together: blue is for the 2SRWB, green is for the Dyck path, and red is for the SRWB.

Theorems & Definitions (21)

  • Lemma 1.1
  • Proposition 1.2: Catalan, see also Larcombe and French LarcombeFrench
  • Corollary 1.3
  • Proposition 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 11 more