content_image/323341.jpg content_image/323342.jpg content_image/323343.jpg content_image/323344.jpg
## 3.3 Proof of Theorem 3.3 

To prove Theorem 3.3, we consider the following decomposition: 

<img src='content_image/323341.jpg'>

It will be shown that n − 1 / $^{2}$M$_{n}$ ( x, t ) = O $_{P}$(1) uniformly in x, t , while d − 1 $_{n,m}$R$_{n}$ ( x, t ) converges in distribution to the limit process in formula (13). Theorem 3.3 then follows because √$_{n}$ = o( d$_{n,m}$ ). 

Martingale part. For fixed x , the following lemma characterizes the asymptotic behav- ior of the martingale part M$_{n}$ ( x, t ). We write 

with ζ$_{j}$ ( x $^{) = 1}${ ψ ( X$_{j}$ ) ≤ x } − E ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } | F j − 1 ) . 

Lemma 3.5. Under the conditions of Theorem 3.3, we have for every x , 

<img src='content_image/323342.jpg'>

<img src='content_image/323343.jpg'>

in D ([0 , 1]) , where B denotes a Brownian motion process and β $^{2}$( x ) = E( ζ 2 $_{1}$( x )) . 

Proof. Define 

<img src='content_image/323344.jpg'>

with X$_{nj}$ ( x ) = n − 1 2 $^{1}${ ψ ( X$_{j}$ ) ≤ x } . In order to show convergence in D ([0 , 1]), we apply the functional martingale central limit theorem as stated in Theorem 18.2 of Billingsley (1999). 

