If E( ψ ( X$_{1}$ ) | F $_{0}$) = 0, it follows by an application of the continuous mapping theorem to the partial sum process 1 √$_{n}$ ∑ ⌊ nt ⌋ j =1 ψ ( X$_{j}$ ) that T$_{n}$ ( τ$_{1}$, τ$_{2}$ ) D −→ T (1 , τ$_{1}$, τ$_{2}$ ), where T (1 , τ$_{1}$, τ$_{2}$ ) corresponds to (15) with Z$_{1}$ denoting a fractional Brownian motion process with Hurst parameter 1 / 2, i.e. under these assumptions Z$_{1}$ denotes a Brownian motion. Note that in this case the limit does not depend on any unknown parameters. The factor σ that appeared in the limit of the partial sum process is canceled out by self-normalization. As an alternative to the self-normalized CUSUM test, Betken (2016b) proposes to use a self-normalized Wilcoxon test for the detection of changes in the mean of long range dependent time series. For the definition of the corresponding test statistic, we consider the ranks defined by R$_{i}$ := rank( X$_{i}$ ) = ∑ n j $^{$_{=1}$1}${ X$_{j}$ ≤ X$_{i}$ } for i = 1 , . . . , n . The self-normalized two-sample Wilcoxon test statistic is then defined by Under the hypothesis of stationarity, the asymptotic distribution of the test statistic is derived in Betken (2016b) for long range dependent subordinated Gaussian processes. Basically, the asymptotics of the test statistic follow from an application of the continuous mapping theorem to the process Theorem 3.3 implies that the above process converges in distribution to the following limit: 1 nd$_{n,m}$ ⌊ nt ⌋ ∑ i =1 n ∑ j = ⌊ nt ⌋ +1 ( $^{1}${ ψ ( X$_{i}$ ) ≤ ψ ( X$_{j}$ ) } − 1 2 ) , t ∈ [0 , 1] . ∫ R J$_{m}$ (Ψ$_{x}$ ◦ σ ) dF$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ ( x ) 1 m ! ( Z$_{m}$ ( t ) − λZ$_{m}$ (1)) , t ∈ [0 , 1] . where S$_{t}$ ( j, k ) := ∑ t h = j ( R$_{h}$ − ¯ R$_{j,k}$ ) with ¯ R$_{j,k}$ := 1 k − j +1 ∑ k t = j R$_{t}$ . The self-normalized $^{Wilcoxon change-point test rejects the hypothesis for large values of max$_{k}$}$∈{⌊ nτ 1 ⌋ ,..., ⌊ nτ 2 ⌋} | G$_{n}$ ( k ) | , where 0 < τ$_{1}$ < τ$_{2}$ < 1. G$_{n}$ ( k ) := ∑ k i $_{=1}$R$_{i}$ − k n ∑ n i $_{=1}$R$_{i}$ { 1 n ∑ k t $_{=1}$S 2 $_{t}$(1 , k ) + 1 n ∑ n t = k $_{+1}$S 2 $_{t}$( k + 1 , n ) } 1 / $_{2}$, 21