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Distance covariance for discretized stochastic processes. (English) Zbl 1462.62356
For an iid sequence of continuous, bounded and measurable stochastic processes $$(X_i,Y_i)$$, $$i=1,2,\ldots$$, on $$[0,1]$$, with generic element $$(X,Y)$$, the authors define the distance covariance $$T_\beta(X,Y)$$ between $$X$$ and $$Y$$ by analogy with the distance covariance between vectors of data, for a parameter $$\beta\in(0,2]$$. They show that $$T_\beta(X,Y)=0$$ if and only if $$X$$ and $$Y$$ are independent. The corresponding sample distance covariance is denoted by $$T_{n,\beta}(X,Y)$$.
In the setting of testing for independence of stochastic processes, assume that we observe our processes $$((X_i,Y_i))_{i=1,\ldots,n}$$ only at discrete time points $$t_0 < \cdots < t_p$$ in $$[0,1]$$. This data is denoted by $$((X_i^{(p)},Y_i^{(p)}))_{i=1,\ldots,n}$$. The main object of study in this paper is the sample distance covariance $$T_{n,\beta}(X^{(p)},Y^{(p)})$$. The authors show that $n\left[T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)\right]$ converges in probability to zero as $$n\to\infty$$ for independent $$X$$ and $$Y$$, provided that $$p\to\infty$$ and the maximal distance between the $$t_i$$ converges to zero sufficiently quickly. They also establish consistency of $$T_{n,\beta}(X^{(p)},Y^{(p)})$$, as well as an asymptotic weighted $$\chi^2$$ distribution for this statistic. Bootstrap procedures are also considered, and simulations are used to illustrate the main results.

##### MSC:
 62H20 Measures of association (correlation, canonical correlation, etc.) 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 60G05 Foundations of stochastic processes 60E10 Characteristic functions; other transforms
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