00-170 Minami N.
On the Poisson Limit Theorems of Sinai and Major (113K, AMSTeX 2.1) Apr 7, 00
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Abstract. Let $f(\varphi)$ be a positive continuous function on $0\leq\varphi\leq\Theta$ , where $\Theta\leq2\pi$ , and let $\xi$ be the number of two-dimensional lattice points in the domain $\Pi_R(f)$ between the curves $r=(R+c_1/R)f(\varphi)$ and $r=(R+c_2/R)f(\varphi)$ , where $c_1<c_2$ are fixed. Randomizing the function $f$ according to a probability law $\bold P$ , and the parameter $R$ according to the uniform distribution $\mu_L$ on the interval $[a_1L,a_2L]$ , Sinai showed that the distribution of $\xi$ under $\bold P\times\mu_L$ converges to a mixture of the Poisson distributions as $L\to\infty$ . Later Major showed that for $\bold P$-almost all $f$ , the distribution of $\xi$ under $\mu_L$ converges to a Poisson distribution as $L\to\infty$ . In this note, we shall give shorter and more transparent proofs to these interesting theorems, at the same time extending the class of $\bold P$ and strengthening the statement of Sinai.

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