Consistency of the Takens estimator for the correlation dimension. (English) Zbl 0928.62072

J. Aaronson et al. [Trans. Am. Math. Soc. 348, No. 7, 2845-2866 (1996; Zbl 0863.60032)] proved consistency results for U-statistics of stationary ergodic, respectively absolutely regular, sequences under the condition that the kernel \(h(x,y)\) is bounded. In this paper, replacing their condition on \( (x,y)\) by a uniform integrability requirement on \(h(X_i,Y_j)\), \(i,j\geq 1\), the authors extend their results. The results obtained enable us to show consistency of the Takens estimator for the correlation dimension, which is a good measure of the complexity of the dynamics.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F05 Central limit and other weak theorems
62G05 Nonparametric estimation
60G10 Stationary stochastic processes


Zbl 0863.60032
Full Text: DOI


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