The goal of this paper is to obtain limiting distributions for the CUSUM and the Wilcoxon test statistic in case of time series that follow the LMSV model. For the observations X$_{1}$, . . . , X$_{n}$ that satisfy the LMSV model, the asymptotic behavior of the partial sum process is described by Theorem 4.10 in Beran et al. (2013) and hence is stated without proof. In order to formulate the result, we introduce the following notation: i.e. F j denotes the σ -field generated by the random variables ε$_{j}$$^{, ε$_{j}$}$− $_{1}$, . . . , η$_{j}$ $^{, η$_{j}$}$− $_{1}$, . . . . Due to this construction, ε$_{j}$ is independent of F j − 1 and Y$_{j}$ is F j − $_{1}$-measurable. Theorem 3.1. Assume that { X$_{j}$, j ≥ 1 } follows the LMSV model. Furthermore, assume that E( ψ $^{2}$( X$_{1}$ )) < ∞ . Define the function Ψ by Ψ( y ) = E ( ψ ( σ ( y ) ε$_{1}$ )) . Denote by m the Hermite rank of Ψ and by J$_{m}$ (Ψ) the corresponding Hermite coefficient. In order to determine the asymptotic behavior of the CUSUM test statistic computed with respect to the observations ψ ( X$_{1}$ ) , . . . , ψ ( X$_{n}$ ), we have to consider the partial sum process ∑ ⌊ nt ⌋ j =1 ( ψ ( X$_{j}$ ) − E ( ψ ( X$_{j}$ ))). For the observations X$_{1}$, . . . , X$_{n}$ that satisfy the LMSV model, the asymptotic behavior 3.1 CUSUM Test for LMSV F j = σ ( ε$_{j}$$^{, ε$_{j}$}$− $_{1}$, . . . , η$_{j}$ $^{, η$_{j}$}$− $_{1}$, . . . ) , 1 √$_{n}$ ⌊ nt ⌋ ∑ j =1 ψ ( X$_{j}$ ) ⇒ σB ( t ) , t ∈ [0 , 1] , 2. If E( ψ ( X$_{1}$ ) | F $_{0}$) = 0 , then 1. If E( ψ ( X$_{1}$ ) | F $_{0}$) = 0 and mD < 1 , then 11 1 d$_{n,m}$ ⌊ nt ⌋ ∑ j =1 ( ψ ( X$_{j}$ ) − E( ψ ( X$_{j}$ ))) ⇒ J$_{m}$ (Ψ) m ! Z$_{m}$ ( t ) , t ∈ [0 , 1] , in D ([0 , 1]) .