2024/10/29 by Timothy Li, Li, Timothy, Shannon Starr +1 · 1 citation
Computer Science · #05A15 #60F10 #60G50 #Combinatorics (math.CO) #Data Management and Algorithms #FOS: Mathematics #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2410.22486
openalex publication_date 2024/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider multifold convolutions of a combinatorial sequence (an)n=0∞: namely, for each k ∈ \N the k-fold convolution is M(k)n(\boldsymbola) = ∑j1+…+jk=n aj1 ⋯ ajk. Let Cn be the Catalan numbers, and let Bn 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 Cn and Bn. But even for combinatorial sequences Bn2 and Bn3, 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.