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Probability Theory and Related Fields

pp 1–42

First Online: 17 May 2017Received: 09 November 2015Revised: 22 February 2017DOI: 10.1007-s00440-017-0781-1

Cite this article as: Chevyrev, I. Probab. Theory Relat. Fields 2017. doi:10.1007-s00440-017-0781-1


We consider random walks and Lévy processes in a homogeneous group G. For all \p > 0\, we completely characterise almost all G-valued Lévy processes whose sample paths have finite p-variation, and give sufficient conditions under which a sequence of G-valued random walks converges in law to a Lévy process in p-variation topology. In the case that G is the free nilpotent Lie group over \\mathbb {R}^d\, so that processes of finite p-variation are identified with rough paths, we demonstrate applications of our results to weak convergence of stochastic flows and provide a Lévy–Khintchine formula for the characteristic function of the signature of a Lévy process. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes.

KeywordsHomogeneous groups Rough paths Lévy processes Random walks Tightness of p-variation Stochastic flows Characteristic functions of signatures The author is supported by a Junior Research Fellowship of St John’s College, Oxford.

Mathematics Subject ClassificationPrimary 60G51 Secondary 60H10 

Author: Ilya Chevyrev

Source: https://link.springer.com/


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