Minami N.
On the Poisson Limit Theorems of Sinai and Major
(113K, AMSTeX 2.1)

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.