Random walks driven by low moment measures

Original · EN

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 → [0,+∞], we introduce a function ΦG,ᵨ which describes the fastest possible decay of n ϕ⁽²ⁿ⁾(e) when ϕis a symmetric continuous probability density such that ∫ρϕ is finite. We estimate ΦG,ᵨ for a variety of groups G and functions ρ. When ρis of the form ρ=ρ∘ δ with ρ:[0,+∞) → [0,+∞), a fixed increasing function, and δ:G → [0,+∞), a natural word length measuring the distance to the identity element in G, ΦG,ᵨ can be thought of as a group invariant.