## 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.

### MSC:

 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
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### References:

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