Israel Journal of Mathematics

, Volume 132, Issue 1, pp 61–107 | Cite as

Classical theorems of probability on Gelfand pairs—Khinchin’s theorems and Cramér’s theorem

  • P. Graczyk
  • C. R. E. Raja


We prove Khinchin’s Theorems for Gelfand pairs (G, K) satisfying a condition (*): (a)G is connected; (b)G is almost connected and Ad (G/M) is almost algebraic for some compact normal subgroupM; (c)G admits a compact open normal subgroup; (d) (G,K) is symmetric andG is 2-root compact; (e)G is a Zariski-connectedp-adic algebraic group; (f) compact extension of unipotent algebraic groups; (g) compact extension of connected nilpotent groups. In fact, condition (*) turns out to be necessary and sufficient forK-biinvariant measures on aforementioned Gelfand pairs to be Hungarian. We also prove that Cramér’s theorem does not hold for a class of Gaussians on compact Gelfand pairs.


Algebraic Group Compact Group Compact Subgroup Maximal Compact Subgroup Gelfand Pair 
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© Hebrew University 2002

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité d’AngersAngers Cedex 01France
  2. 2.Département de MathématiquesIndian Statistical InstituteBangaloreIndia

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