From the introduction: We assume that \(n\) observations \(X_1, \dots,X_n\) can be written as \[ X_i=\begin{cases} \mu+\sigma e_i,\quad & 1\leq i\leq k^*,\\ \mu+\tau e_i, \quad & k^*<i\leq n,\end{cases} \] where \(\mu, \sigma,\tau\) and \(k^*\) are unknown and \(e_i=\sum^\infty_{j=1}a_j \varepsilon_{i-j}\), where \[ a_j\sim c_0j^{-(1+\alpha) /2},\;(j\to\infty), \;0<\alpha<1,\;\sum^\infty_{j=1}\alpha_j^2=1,\tag{1} \] \(\{ \varepsilon_i\}\) is an i.i.d. process with \(E\varepsilon_1=0\), \(E \varepsilon_1^2 =1\). The last equality of (1) is to ensure that the variance of \(e_i\) is 1. The symbol “\(\sim\)” indicates that the ratio of left- and right-hand sides tends to one. Assuming that \(\sigma\neq\tau\), our goal is to investigate whether the variance of the observations has changed at an unknown time. That is, we want to test the null hypothesis \(H_0:k^*\geq n\) against the alternative \(H_A:1\leq k^*<n.\)