Let $(X_k)_{ k\geq1}$ be a Gaussian long-range dependent process with $EX_1=0$, $EX_1^2=1$ and covariance function $r(k)=k^{ -D}L(k)$. For any measurable function $G$ let $(Y_k)_{ k\geq1}=(G(X_k))_{ k\geq1}$. We study the asymptotic behaviour of the associated sequential empirical process $\left(R_N(x,t)\right)$ with respect to a weighted norm $\|\cdot\|_w$. We show that, after an appropriate normalization, $\left(R_N(x,t)\right)$ converges weakly in the space of c\`adl\`ag functions with finite weighted norm to a Hermite process.
Categories: math.PR math.ST stat.TH