### Abstract

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 Π_{R}(f) between the curves r = (R + c_{1}/R)f(φ) and r = (R + c_{2}/R)f(φ), where c_{1} < c_{2} are fixed. Randomizing the function f according to a probability law P, and the parameter R according to the uniform distribution μ_{L} on the interval [a_{1}L, a_{2}L], Sinai showed that the distribution of ξ under P × μ_{L} converges to a mixture of the Poisson distributions as L → ∞. Later Major showed that for P-almost all f, the distribution of ξ under μ_{L} 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 language | English |
---|---|

Pages (from-to) | 203-247 |

Number of pages | 45 |

Journal | Communications in Mathematical Physics |

Volume | 213 |

Issue number | 1 |

Publication status | Published - 2000 Sep |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Communications in Mathematical Physics*,

*213*(1), 203-247.

**On the poisson limit theorems of Sinai and major.** / Minami, Nariyuki.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 213, no. 1, pp. 203-247.

}

TY - JOUR

T1 - On the poisson limit theorems of Sinai and major

AU - Minami, Nariyuki

PY - 2000/9

Y1 - 2000/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034349952&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034349952&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034349952

VL - 213

SP - 203

EP - 247

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -