On the poisson limit theorems of Sinai and major

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Let f(φ) be a positive continuous function on 0 ≤ φ ≤ Θ, where Θ ≤ 2π, and let ξ be the number of two-dimensional lattice points in the domain ΠR(f) between the curves r = (R + c1/R)f(φ) and r = (R + c2/R)f(φ), where c1 < c2 are fixed. Randomizing the function f according to a probability law P, and the parameter R according to the uniform distribution μL on the interval [a1L, a2L], Sinai showed that the distribution of ξ under P × μL converges to a mixture of the Poisson distributions as L → ∞. Later Major showed that for P-almost all f, the distribution of ξ under μL converges to a Poisson distribution as L → ∞. In this note, we shall give shorter and more transparent proofs to these interesting theorems, at the same time extending the class of P and strengthening the statement of Sinai.

Original languageEnglish
Pages (from-to)203-247
Number of pages45
JournalCommunications in Mathematical Physics
Issue number1
Publication statusPublished - 2000 Sept
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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