If { ε$_{j}$, j ≥ 1 } are centered two-sided Pareto distributed with parameter α , it follows by some basic calculations that where µ is the expected value of ˜ ε$_{1}$ , i.e. µ := α α − $_{1}$, and As a result, we have Since our theoretical results do not provide a non-degenerate asymptotic distribution for the Wilcoxon test statistics, we only consider CUSUM and self-normalized CUSUM test for the first scenario (when the ε$_{j}$ , j ≥ 1, are standard normally distributed). Note that in this case the size of the non-self-normalized CUSUM change point test Table 1: Rejection rates of the CUSUM tests for LMSV time series (standard normal ε$_{j}$, j ≥ 1) of length n with Hurst parameter H and a shift in the mean of height h after a proportion τ = 0 . 5. The calculations are based on 5,000 simulation runs. f$_{X}$ ( x ) = 1 x ∫ ∞ µ h$_{α}$ ( x, u ) du $^{1}${ x> 0 } − 1 x ∫ µ 1 h$_{α}$ ( x, u ) du $^{1}${ x< 0 } , J$_{1}$ (Ψ$_{x}$ ◦ σ ) = − ∫ ∞ µ h$_{α}$ ( x, u ) du $^{1}${ x> 0 } + ∫ µ 1 h$_{α}$ ( x, u ) du $^{1}${ x< 0 } , h$_{α}$ ( x, u ) = $_{αu}$− α − $^{1}$ϕ ( 1 2 [ log ( ( u − µ ) $^{2}$) − log( x $^{2}$) ] ) $^{1}${ u> 1 } . ∫ J$_{1}$ (Ψ$_{x}$ ◦ σ ) dF$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ ( x ) = ∫ ∞ 0 1 x { (∫ µ 1 h$_{α}$ ( x, u ) du ) 2 − (∫ ∞ µ h$_{α}$ ( x, u ) du ) 2 } dx. CUSUM self-norm. CUSUM H n h = 0 h = 0 . 5 h = 1 h = 2 h = 0 h = 0 . 5 h = 1 h = 2 0.6 500 0.046 0.384 0.958 1.000 0.043 0.372 0.853 0.998 1000 0.045 0.728 1.000 1.000 0.047 0.627 0.975 1.000 2000 0.043 0.958 1.000 1.000 0.044 0.856 0.998 1.000 0.7 500 0.064 0.392 0.962 1.000 0.044 0.395 0.858 0.995 1000 0.056 0.736 0.999 1.000 0.044 0.629 0.974 1.000 2000 0.050 0.964 1.000 1.000 0.046 0.869 0.997 1.000 0.8 500 0.074 0.369 0.969 1.000 0.039 0.462 0.860 0.993 1000 0.074 0.734 1.000 1.000 0.044 0.670 0.963 0.999 2000 0.072 0.966 1.000 1.000 0.048 0.851 0.996 1.000 0.9 500 0.095 0.311 0.977 1.000 0.045 0.583 0.875 0.980 1000 0.101 0.739 0.998 1.000 0.045 0.736 0.946 0.996 2000 0.101 0.979 1.000 1.000 0.043 0.860 0.987 1.000 24