  Repeated convolution of a probability measure on Z leads to the central limit
theorem and other limit theorems. This paper investigates what kinds of results
remain without positivity. It reviews theorems due to Schoenberg, Greville, and
Thomee which are motivated by applications to data smoothing (Schoenberg and
Greville) and finite difference schemes (Thomee). Using Fourier transform
arguments, we prove detailed decay bounds for convolution powers of finitely
supported complex functions on Z. If M is an hermitian contraction, an estimate
for the off-diagonal entries of the powers M^n k of M^k = I -(I-M)^k is
obtained. This generalizes the Carne-Varopoulos Markov chain estimate.
