The empirical process of some long-range dependent sequences with an application to U-statistics
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Published version
Date
1989-12-01
DOI
Authors
Dehling, Herold
Taqqu, Murad S.
Version
OA Version
Citation
H DEHLING, MS TAQQU. 1989. "The Empirical Process of some Long-Range Dependent Sequences with an Application to U-Statistics ." The Annals of Statistics, Volume 17, Issue 4, pp. 1767 - 1783 (17). https://doi.org/10.1214/aos/1176347394
Abstract
Let (Xj)∞ j = 1 be a stationary, mean-zero Gaussian process with covariances r(k) = EXk + 1 X1 satisfying r(0) = 1 and r(k) = k-DL(k) where D is small and L is slowly varying at infinity. Consider the two-parameter empirical process for G(Xj), $\bigg\{F_N(x, t) = \frac{1}{N} \sum^{\lbrack Nt \rbrack}_{j = 1} \lbrack 1\{G(X_j) \leq x\} - P(G(X_1) \leq x) \rbrack; // -\infty < x < + \infty, 0 \leq t \leq 1\bigg\},$ where G is any measurable function. Noncentral limit theorems are obtained for FN(x, t) and they are used to derive the asymptotic behavior of some suitably normalized von Mises statistics and U-statistics based on the G(Xj)'s. The limiting processes are structurally different from those encountered in the i.i.d. case.
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© 1989 Institute of Mathematical Statistics