Random walks driven by low moment measures

• Autores: Alexander Bendikov, Laurent Saloff-Coste
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 40, Nº. 6, 2012, págs. 2539-2588
• Idioma: inglés
• DOI: 10.1214/11-AOP687
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• Resumen

  • 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 ϕ is a symmetric continuous probability density such that ∫ϱϕ is finite. We estimate ΦG,ϱ for a variety of groups G and functions ϱ. When ϱ is 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.