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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 

<img src='content_image/310855.jpg'>

<img src='content_image/310856.jpg'>

<img src='content_image/310857.jpg'>

<img src='content_image/310858.jpg'>

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). 

<img src='content_image/310859.jpg'>

Note that in this case the size of the non-self-normalized CUSUM change point test 

