content_image/303468.jpg content_image/303473.jpg content_image/303469.jpg
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. 

## 3.1 CUSUM Test for LMSV 

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 

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: 

<img src='content_image/303468.jpg'>

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. 

1. If E( ψ ( X$_{1}$ ) | F $_{0}$) ̸ = 0 and mD < 1 , then 

<img src='content_image/303473.jpg'>

2. If E( ψ ( X$_{1}$ ) | F $_{0}$) = 0 , then 

<img src='content_image/303469.jpg'>

