4.3 Change in the tail index To test for a change in the tail index, we choose ψ ( x ) = log( x $^{2}$). CUSUM: In this case, E( ψ ( X$_{1}$ ) | F $_{0}$) = 0 and hence long memory affects the limiting distribution of the CUSUM statistic. Moreover, so that the Hermite rank of Ψ equals the Hermite rank of h = log ◦ σ . We note further that if ψ ( x ) = log( x $^{2}$), we have The first summand converges to 2( J$_{m}$ (log ◦ σ ) /m !) Z$_{m}$ ( t ) as a consequence of the functional non-central limit theorem discussed in Section 2.1. The second term is o$_{P}$ (1) uniformly in t by Donsker’s theorem, if Var(log ε 2 $_{1}$) < ∞ . As a result, this observation is consistent with Corollary 3.2. Wilcoxon: Due to monotonicity of the logarithm, choosing ψ ( x ) = log( x $^{2}$) yields the same test statistic as if testing for a change in variance, i.e. if we choose ψ ( x ) = x $^{2}$. Moreover, the Wilcoxon test statistic is robust, i.e. it is designed to reduce the effect of heavy tails, so that it does not seem advisable to apply Wilcoxon-like tests when testing for a change in the tail index. 5 Self-normalization An application of the CUSUM test presupposes knowledge of the normalizing sequence d$_{n,m}$ (if E( ψ ( X$_{1}$ ) | F $_{0}$) = 0) and of the coefficients J$_{m}$ (Ψ) or σ that appear in the limit of 19 J$_{m}$ (Ψ) = 2 ∫ log( σ ( z )) H$_{m}$ ( z ) ϕ ( z ) dz = 2 J$_{m}$ (log ◦ σ ) , 1 d$_{n,m}$ ⌊ nt ⌋ ∑ j =1 ( log ( X 2 j ) − E log ( X 2 j )) = 2 d$_{n,m}$ ⌊ nt ⌋ ∑ j =1 (log σ ( Y$_{j}$ ) − E log σ ( Y$_{j}$ )) + √$_{n}$ d$_{n,m}$ 1 √$_{n}$ ⌊ nt ⌋ ∑ j =1 ( log ( ε 2 j ) − E log ( ε 2 j )) .