content_image/320850.jpg content_image/320851.jpg content_image/320852.jpg content_image/320849.jpg content_image/320848.jpg content_image/320847.jpg
By Corollary 1, x S$_{p}$ − 1 = x p − S$_{p}$ p + o ( p − $^{1}$) a.s.. Using this fact, after some algebra, we get 

and 

<img src='content_image/320850.jpg'>

<img src='content_image/320851.jpg'>

<img src='content_image/320852.jpg'>

It follows that 

<img src='content_image/320849.jpg'>

where the last equality in (55) follows from (3) and Proposition 2. 

The last equality in (55), (4) and the fact that 

<img src='content_image/320848.jpg'>

imply that 

<img src='content_image/320847.jpg'>

which establishes (11). The rest of the statements of Theorem 1 follow from (10), (11), and Lemmas 12 and A2 of OMH. □ 

