The paper is structured as follows: in Section 2 we collect some results on subordinated Gaussian processes. In Section 3 we discuss the change-point problem. In particular, we consider CUSUM and Wilcoxon test. The asymptotic behavior of the former is a direct consequence of the existing results, while the latter requires a new theorem (see Theorem 3.3) on the limiting behavior of empirical processes based on LMSV data. Section 4 is devoted to examples in case of testing for changes in the mean, in the variance and in the tail. Since the test statistics and/or limiting distributions involve unknown quantities, self-normalization is considered in Section 5. In fact, in Section S2 we perform simulation studies and indicate that self-normalization provides robustness. 2 Preliminaries 2.1 Some properties of subordinated Gaussian sequences The main tool for studying the asymptotic behavior of subordinated Gaussian sequences is the Hermite expansion. For a stationary, long range dependent Gaussian process { Y$_{j}$, j ≥ 1 } and a measurable function g such that E ( g $^{2}$( Y$_{1}$ )) < ∞ the corresponding Hermite expansion is defined by where H$_{q}$ is the q -th Hermite polynomial, 7 g ( Y$_{1}$ ) − E ( g ( Y$_{1}$ )) = ∞ ∑ q = m J$_{q}$ ( g ) q ! H$_{q}$ ( Y$_{1}$ ) , J$_{q}$ ( g ) = E ( g ( Y$_{1}$ ) H$_{q}$ ( Y$_{1}$ )) C: CUSUM and Wilcoxon tests for a change in the tail index of LMSV models are typically affected by long memory; see Section 4.3