content_image/315849.jpg content_image/315850.jpg content_image/315851.jpg content_image/315852.jpg content_image/315853.jpg
{ X$_{j}$, j ≥ 1 } , i.e. our preliminary goal is to prove a limit theorem for the two-parameter empirical process 

<img src='content_image/315849.jpg'>

where now F$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ denotes the distribution function of ψ ( X$_{1}$ ) with X$_{1}$ = σ ( Y$_{1}$ ) ε$_{1}$ . To state the weak convergence, we introduce the following notation: 

<img src='content_image/315850.jpg'>

Theorem 3.3. Assume that { X$_{j}$, j ≥ 1 } follows the LMSV model. Moreover, assume that 

<img src='content_image/315851.jpg'>

Let m denote the Hermite rank of the class { $^{1}${ σ ( Y 1 ) ≤ x } − F$_{σ}$$_{(}$$_{Y}$$_{1}$$_{)}$ ( x ) , x ∈ R } . If mD < 1 , then 

<img src='content_image/315852.jpg'>

in D ([ −∞ , ∞ ] × [0 , 1]) . 

The proof of this theorem is given in Section 3.3. At this moment, we derive the asymptotic distribution of the Wilcoxon statistics. 

Corollary 3.4. Under the conditions of Theorem 3.3 

<img src='content_image/315853.jpg'>

Proof of Corollary 3.4. According to Dehling et al. (2013), the asymptotic distribution of the Wilcoxon test statistic can be derived directly from the limit of the two-parameter empirical process if the sequence { X$_{j}$, j ≥ 1 } is ergodic. Ergodicity is obvious, since X$_{j}$ can be represented as a measurable function of the i.i.d. vectors { ( η$_{j}$, ε$_{j}$ ) , j ≥ 1 } . 

