Multifold Convolutions, Generating Functions and 1d Random Walks

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.