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. where B denotes a Brownian motion process and σ 2 = E( ψ $^{2}$( X$_{1}$ )) . It is important to note that the Hermite rank of Ψ does not necessarily correspond to the Hermite rank of σ ; see Section 4. 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 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 12 ⌊ nt ⌋ ∑ j =1 ( $^{1}${ g ( Y$_{j}$ ) ≤ x } − F$_{g}$$_{(}$$_{Y}$$_{1}$$_{)}$ ( x ) ) , x ∈ [ −∞ , ∞ ] , t ∈ [0 , 1] , 1 √$_{n}$ sup 0 ≤ λ ≤ 1 C$_{n}$ ( λ ) ⇒ σ sup 0 ≤ t ≤ 1 | B ( t ) − tB (1) | , 1 d$_{n,m}$ sup 0 ≤ λ ≤ 1 C$_{n}$ ( λ ) ⇒ | J$_{m}$ (Ψ) | m ! sup 0 ≤ t ≤ $_{1}$| Z$_{m}$ ( t ) − tZ$_{m}$ (1) | . (11) 1. If E( ψ ( X$_{1}$ ) | F $_{0}$) = 0 and mD < 1 , then 2. If E( ψ ( X$_{1}$ ) | F $_{0}$) = 0 3.2 Wilcoxon test for LMSV