content_image/310843.jpg content_image/310842.jpg content_image/310847.jpg
Pareto distributed, i.e. for some α, c > 0 f$_{ε}$ ( x ) = αc α x α +1 $^{1}${ x ≥ c } , then, as a result, 

<img src='content_image/310843.jpg'>

Hence, J$_{1}$ (Ψ$_{x}$ ◦ σ ) ̸ = 0. In any case, long memory in LMSV influences the asymptotic behavior of the Wilcoxon statistic. 

## 4.2 Change in the variance 

To test for a change in the variance, we choose ψ ( x ) = x $^{2}$. 

CUSUM: Recall again that the function Ψ in Theorem 3.1 is defined as Ψ( y ) = E ( ψ ( σ ( y ) ε$_{1}$ )). Then E( ψ ( X$_{1}$ ) | F $_{0}$) ̸ = 0 and hence long memory affects the limiting behavior of the CUSUM statistic. Moreover, 

<img src='content_image/310842.jpg'>

i.e. the Hermite rank of Ψ equals the Hermite rank of σ $^{2}$. If mD < 1, then the limiting behavior of the CUSUM statistic is described by (11). Hence, long memory in LMSV influences the asymptotic behavior of the CUSUM statistic when testing for a change in the variance. 

Wilcoxon: Recall again that Ψ$_{x}$( y ) = P ( ψ ( yε$_{1}$ ) ≤ x ). For x ≥ 0, we have 

<img src='content_image/310847.jpg'>

If σ ( y ) = exp( y ), then we can conclude by the same argument as in the case of testing for a change in the mean that J$_{1}$ (Ψ$_{x}$ ◦ σ ) ̸ = 0. Hence, long memory in LMSV time series influences the asymptotic behavior of the Wilcoxon statistic when testing for a change in the variance. 

