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Random walks driven by low moment measures

We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group G and any positive function $ρ:G \rightarrow [0,+\infty]$, we introduce a function $Φ_{G,ρ}$ which describes the fastest possible decay of $n \mapsto ϕ^{(2n)}(e)$ when ϕis a symmetric continuous probability density such that $\intρϕ$ is finite. We estimate $Φ_{G,ρ}$ for a variety of groups G and functions ρ. When ρis of the form $ρ=ρ\circ δ$ with $ρ:[0,+\infty) \rightarrow [0,+\infty)$, a fixed increasing function, and $δ:G \rightarrow [0,+\infty)$, a natural word length measuring the distance to the identity element in G, $Φ_{G,ρ}$ can be thought of as a group invariant.