  The basic aim of this paper is to study asymptotic properties of the
convolution powers K^(n) = K * K * ... * K of a possibly non-symmetric
probability density K on a locally compact, compactly generated group G. If K
is centered, we show that the Markov operator T associated with K is analytic
in L^p(G) for 1<p<\infty, and establish Davies-Gaffney estimates in L^2 for the
iterated operators T^n. These results enable us to obtain various Gaussian
bounds on K^(n). In particular, when G is a Lie group we recover and extend
some estimates of Alexopoulos and of Varopoulos for convolution powers of
centered densities and for the heat kernels of centered sublaplacians. Finally,
in case G is amenable, we discover that the properties of analyticity or
Davies-Gaffney estimates hold only if K is centered.
