On the poisson limit theorems of Sinai and major

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 + 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 μ_L on the interval [a₁L, a₂L], 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.