Abstract
A local limit theorem is proven on connected, simply connected nilpotent Lie groups, for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization. The result extends an earlier local limit theorem of Alexopoulos which treated absolutely continuous measures with a continuous density of compact support, and also extends local limit theorems of Breuillard and Diaconis–Hough which treated general measures on the Heisenberg group.
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Robert Hough is supported by NSF Grant DMS-1712682, “Probabilistic methods in discrete structures and applications.”
The author thanks Persi Diaconis for his continued interest in the project, and Terence Tao and Emmanuel Breuillard for helpful comments.
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Hough, R. The local limit theorem on nilpotent Lie groups. Probab. Theory Relat. Fields 174, 761–786 (2019). https://doi.org/10.1007/s00440-018-0864-7
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DOI: https://doi.org/10.1007/s00440-018-0864-7
Keywords
- Random walk on a group
- Local limit theorem
- Nilpotent group
Mathematics Subject Classification
- Primary 60F05
- 60B15
- 20B25
- 22E25
- 60J10
- 60E10
- 60F25
- 60G42