On the Poisson Limit Theorems of Sinai and Major |
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Authors: | Nariyuki Minami |
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Institution: | Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan.?E-mail: minami'sakura.cc.tsukuba.ac.jp, JP
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Abstract: | Let f(ϕ) be a positive continuous function on 0 ≤ϕ≤Θ, where Θ≤ 2 π, 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.
Received: 15 June 1999 / Accepted: 11 February 2000 |
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Keywords: | |
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