content_image/301731.jpg content_image/301732.jpg content_image/301733.jpg
Using the reduction principle (7) with g = σ and the integrability condition (12), we con- clude that the first summand converges to 0 in probability, uniformly in x, t . Furthermore, 

<img src='content_image/301731.jpg'>

Denoting by ϕ the standard normal density, integration by parts yields 

The proof is concluded by (6). 

## 4 Examples 

## 4.1 Change in the mean 

To test for a change in the mean, we choose ψ ( x ) = x . 

<img src='content_image/301732.jpg'>

CUSUM: Recall that the function Ψ in Theorem 3.1 is defined as Ψ( y ) = E ( ψ ( σ ( y ) ε$_{1}$ )). In this case, E( ψ ( X$_{1}$ ) | F $_{0}$) = 0. Therefore, the CUSUM statistic converges to a Brownian bridge. Hence, long memory in LMSV does not influence the asymptotic behavior of the CUSUM statistic when testing for a change in the mean. 

Wilcoxon: Recall that Ψ$_{x}$( y ) = P ( ψ ( yε$_{1}$ ) ≤ x ). Using integration by parts and noting that ( d/dz ) ϕ ( z ) = − zϕ ( z ), we conclude 

<img src='content_image/301733.jpg'>

where f$_{ε}$ is the density of ε$_{1}$ (if it exists). Here, different scenarios are possible: if σ ( y ) = y $^{2}$, then the integrand is antisymmetric in z for any choice of f$_{ε}$ . Hence, J$_{1}$ (Ψ$_{x}$ ◦ σ ) = 0 and some calculations show that the Hermite rank of Ψ$_{x}$ ◦ σ is 2. If σ ( y ) = exp( y ) and ε$_{1}$ is 

