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We study the decay of convolution powers of a large family $\mu_{S,a}$ ofmeasures on finitely generated nilpotent groups. Here, $S=(s_1,...,s_k)$ is agenerating $k$-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 givesprobability 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 ofreturn $\mu_{S,a}^{(n)}(e)$ as $n$ tends to infinity. This behavior depends insomewhat subtle ways on interactions between the $k$-tuple $a$ and thepositions of the generators $s_i$ within the lower central series$G_{j}=[G_{j-1},G]$, $G_1=G$.

Random walks on nilpotent groups driven by measures supported on powers of generators