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The local limit theorem on nilpotent Lie groups

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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Authors and Affiliations

Authors

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Correspondence to Robert Hough.

Additional information

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