{ X$_{j}$, j ≥ 1 } , i.e. our preliminary goal is to prove a limit theorem for the two-parameter empirical process 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: Theorem 3.3. Assume that { X$_{j}$, j ≥ 1 } follows the LMSV model. Moreover, assume that 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 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. 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 } . Corollary 3.4. Under the conditions of Theorem 3.3 G$_{n}$ ( x, t ) = ⌊ nt ⌋ ∑ j =1 ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } − F$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ ( x ) ) , Ψ$_{x}$( y ) = P ( ψ ( yε$_{1}$ ) ≤ x ) . ∫ d du P ( ψ ( uε$_{1}$ ) ≤ x ) du < ∞ . (12) 1 d$_{n,m}$ G$_{n}$ ( x, t ) ⇒ J$_{m}$ (Ψ$_{x}$ ◦ σ ) m ! Z$_{m}$ ( t ) , x ∈ [ −∞ , ∞ ] , t ∈ [0 , 1] , (13) 1 nd$_{n,m}$ sup λ ∈ [0 , 1] W$_{n}$ ( λ ) ⇒ ∣ ∫ J$_{m}$ (Ψ$_{x}$ ◦ σ ) dF$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ ( x ) ∣ 1 m ! sup λ ∈ [0 , 1] | Z$_{m}$ ( λ ) − λZ$_{m}$ (1) | . 13