3.3 Proof of Theorem 3.3 To prove Theorem 3.3, we consider the following decomposition: 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 , 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). G$_{n}$ ( x, t ) = ⌊ nt ⌋ ∑ j =1 ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } − E ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } | F j − 1 )) + ⌊ nt ⌋ ∑ j =1 ( E ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } | F j − 1 ) − F$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ ( x ) ) =: M$_{n}$ ( x, t ) + R$_{n}$ ( x, t ) . M$_{n}$ ( t ) := M$_{n}$ ( x, t ) = ⌊ nt ⌋ ∑ j =1 ζ$_{j}$ ( x ) 1 √$_{n}$ M$_{n}$ ( t ) ⇒ β ( x ) B ( t ) , t ∈ [0 , 1] , ζ$_{nj}$ = $_{n}$− 1 2 ζ$_{j}$ ( x ) = X$_{nj}$ ( x ) − E( X$_{nj}$ ( x ) | F j − $_{1}$) 14 Proof. Define in D ([0 , 1]) , where B denotes a Brownian motion process and β $^{2}$( x ) = E( ζ 2 $_{1}$( x )) .