Weak Convergence of the Sequential Empirical Process of some Long-Range Dependent Sequences with Respect to a Weighted Norm
Abstract
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 càdlàg functions with finite weighted norm to a Hermite process.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2013
- DOI:
- 10.48550/arXiv.1312.5894
- arXiv:
- arXiv:1312.5894
- Bibcode:
- 2013arXiv1312.5894B
- Keywords:
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- Mathematics - Probability;
- Mathematics - Statistics Theory
- E-Print:
- Statistics and Probability Letters (2015), 96, pp. 170-179