  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. Following previous work of P. Diaconis
and the authors, we explore the more general setting in which $\phi$ takes on
complex values. This problem, originally motivated by the problem of Erastus L.
De Forest in data smoothing, has found applications to the theory of stability
of numerical difference schemes in partial differential equations. For a
complex valued function $\phi$ on $\mathbb{Z}^d$, we ask and address four basic
and fundamental questions about the convolution powers $\phi^{(n)}$ which
concern sup-norm estimates, generalized local limit theorems, pointwise
estimates, and stability. This work extends one-dimensional results of I. J.
Schoenberg, T. N. E. Greville, P. Diaconis and the second author and, in the
context of stability theory, results by V. Thom\'ee and M. V. Fedoryuk.
