  In this paper we study the behaviour of convolution powers of probability
measures $\mu$ on $\Z$, such that $(\mu(n))_{n\in \N}$ is completely monotone
or such that $\nu$ is centered with a second moment. In particular we exhibit
many new examples of probability measures on $\Z$ having the so called Ritt
property and whose convolution powers satisfy weak type maximal inequalities in
$\ell^1(\Z)$.
