Convolution powers of complex-valued functions on \(\mathbb{Z}^d\)

The study of convolution powers of a finitely supported probability distribution \(\phi\) on the \(d\)-dimensional square lattice is central to random walk theory. For instance, the \(n\)th convolution power \(\phi^{(n)}\) is the distribution of the \(n\)th step of the associated random walk and is described by the classical local limit theorem. When such distributions take on complex values, their convolution powers exhibit surprising and disparate behaviors not seen in the probabilistic setting. In this talk, I will discuss new results concerning the asymptotic behavior of convolution powers of complex-valued functions on \(\mathbb{Z}^d\), specifically generalized local limit theorems and sup-norm estimates. This joint work with Laurent Saloff-Coste extends previous results by I. J. Shoenberg, T. N. E. Greville, P. Diaconis and L. Saloff-Coste.