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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.

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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