  We study the decay of convolution powers of a large family $\mu_{S,a}$ of
measures on finitely generated nilpotent groups. Here, $S=(s_1,...,s_k)$ is a
generating $k$-tuple of group elements and $a= (\alpha_1,...,\alpha_k)$ is a
$k$-tuple of reals in the interval $(0,2)$. The symmetric measure $\mu_{S,a}$
is supported by $S^*=\{s_i^{m}, 1\le i\le k,\,m\in \mathbb Z\}$ and gives
probability proportional to $$(1+m)^{-\alpha_i-1}$$ to $s_i^{\pm m}$,
$i=1,...,k,$ $m\in \mathbb N$. We determine the behavior of the probability of
return $\mu_{S,a}^{(n)}(e)$ as $n$ tends to infinity. This behavior depends in
somewhat subtle ways on interactions between the $k$-tuple $a$ and the
positions of the generators $s_i$ within the lower central series
$G_{j}=[G_{j-1},G]$, $G_1=G$.
