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Distance covariance in metric spaces. (English) Zbl 1292.62087

Ann. Probab. 41, No. 5, 3284-3305 (2013); errata ibid. 46, No. 4, 2400-2405 (2018); second errata ibid. 49, No. 5, 2668-2670 (2021).
Summary: We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Székely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.

MSC:

62H20 Measures of association (correlation, canonical correlation, etc.)
62H15 Hypothesis testing in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
60B11 Probability theory on linear topological spaces

Software:

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References:

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