content_image/313514.jpg content_image/313513.jpg content_image/313512.jpg
in D ([0 , 1]) , where B denotes a Brownian motion process and σ 2 = E( ψ $^{2}$( X$_{1}$ )) . 

As an immediate consequence of Theorem 3.1, we obtain the asymptotic distribution of the CUSUM statistic. 

Corollary 3.2. Assume that the assumptions of Theorem 3.1 hold. 

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

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

where B denotes a Brownian motion process and σ 2 = E( ψ $^{2}$( X$_{1}$ )) . 

<img src='content_image/313514.jpg'>

<img src='content_image/313513.jpg'>

It is important to note that the Hermite rank of Ψ does not necessarily correspond to the Hermite rank of σ ; see Section 4. 

## 3.2 Wilcoxon test for LMSV 

For subordinated Gaussian time series { g ( Y$_{j}$ ) , j ≥ 1 } , where { Y$_{j}$, j ≥ 1 } is a stationary Gaussian LRD process and g is a measurable function, the asymptotic distribution of the Wilcoxon test statistic W$_{n}$ is derived from the limiting behavior of the two-parameter empirical process 

<img src='content_image/313512.jpg'>

where F$_{g}$$_{(}$$_{Y}$$_{1}$$_{)}$ denotes the distribution function of g ( Y$_{1}$ ); see Dehling et al. (2013). 

In order to determine the asymptotic distribution of the Wilcoxon test statistic for the LMSV model, we need to establish an analogous result for the stochastic volatility process 

