On distance covariance in metric and Hilbert spaces. (English) Zbl 1470.62073
Summary: Distance covariance is a measure of dependence between two random variables that take values in two, in general different, metric spaces, see [G. J. Székely et al., Ann. Stat. 35, No. 6, 2769–2794 (2007; Zbl 1129.62059)] and [R. Lyons, Ann. Probab. 41, No. 5, 3284–3305 (2013; Zbl 1292.62087)]. It is known that the distance covariance, and its generalization \(\alpha\)-distance covariance, can be defined in several different ways that are equivalent under some moment conditions. The present paper considers four such definitions and find minimal moment conditions for each of them, together with some partial results when these conditions are not satisfied.
Another purpose of the present paper is to improve existing results on consistency of distance covariance, estimated using the empirical distribution of a sample. The paper also studies the special case when the variables are Hilbert space valued, and shows under weak moment conditions that two such variables are independent if and only if their (\(\alpha\)-)distance covariance is 0; this extends results by Lyons [loc. cit.] and H. Dehling et al. [Bernoulli 26, No. 4, 2758–2789 (2020; Zbl 1462.62356)]. The proof uses a new definition of distance covariance in the Hilbert space case, generalizing the definition for Euclidean spaces using characteristic functions by Székely et al. [loc. cit.].
Another purpose of the present paper is to improve existing results on consistency of distance covariance, estimated using the empirical distribution of a sample. The paper also studies the special case when the variables are Hilbert space valued, and shows under weak moment conditions that two such variables are independent if and only if their (\(\alpha\)-)distance covariance is 0; this extends results by Lyons [loc. cit.] and H. Dehling et al. [Bernoulli 26, No. 4, 2758–2789 (2020; Zbl 1462.62356)]. The proof uses a new definition of distance covariance in the Hilbert space case, generalizing the definition for Euclidean spaces using characteristic functions by Székely et al. [loc. cit.].
MSC:
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62R20 | Statistics on metric spaces |
| 60B11 | Probability theory on linear topological spaces |
Software:
DLMFReferences:
| [1] | Baker, C. R. Joint measures and cross-covariance operators. Trans. Amer. Math. Soc., 186, 273-289 (1973). MR336795. · Zbl 0304.28008 |
| [2] | Bennett, C. and Sharpley, R. Interpolation of operators, volume 129 of Pure and Applied Mathe-matics. Academic Press, Inc., Boston, MA (1988). ISBN 0-12-088730-4. MR928802. · Zbl 0647.46057 |
| [3] | Berg, C., Christensen, J. P. R., and Ressel, P. Harmonic analysis on semigroups. Theory of positive definite and related functions, volume 100 of Graduate Texts in Mathematics. Springer-Verlag, New York (1984). ISBN 0-387-90925-7. MR747302. · Zbl 0619.43001 |
| [4] | Bergh, J. and Löfström, J. Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York (1976). MR0482275. · Zbl 0344.46071 |
| [5] | Billingsley, P. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney (1968). MR0233396. · Zbl 0172.21201 |
| [6] | Bogachev, V. I. and Kolesnikov, A. V. The Monge-Kantorovich problem: achievements, connections, and prospects. Uspekhi Mat. Nauk, 67 (5(407)), 3-110 (2012). MR3058744. English translation: Russian Math. Surveys 67 (2012), no. 5, 785-890. · Zbl 1276.28029 |
| [7] | Conway, J. B. A course in functional analysis, volume 96 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition (1990). ISBN 0-387-97245-5. MR1070713. · Zbl 0706.46003 |
| [8] | Dehling, H., Matsui, M., Mikosch, T., Samorodnitsky, G., and Tafakori, L. Distance covariance for discretized stochastic processes. Bernoulli, 26 (4), 2758-2789 (2020). MR4140528. · Zbl 1462.62356 |
| [9] | Feuerverger, A. A consistent test for bivariate dependence. Int. Stat. Rev., 61 (2), 419-433 (1993). MR4205649. · Zbl 1462.62356 |
| [10] | Gretton, A., Bousquet, O., Smola, A., and Schölkopf, B. Measuring statistical dependence with Hilbert-Schmidt norms. In Algorithmic learning theory, volume 3734 of Lecture Notes in Comput. Sci., pp. 63-77. Springer, Berlin (2005). MR2255909. · Zbl 0826.62032 |
| [11] | Gut, A. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, second edition (2013). ISBN 978-1-4614-4707-8; 978-1-4614-4708-5. MR2977961. · Zbl 1168.62354 |
| [12] | Jakobsen, M. E. Distance covariance in metric spaces: non-parametric independence testing in metric spaces (2017). Master’s thesis. arXiv: 1706.03490v1. · Zbl 1267.60001 |
| [13] | Janson, S. Gaussian Hilbert spaces, volume 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1997). ISBN 0-521-56128-0. MR1474726. · Zbl 0887.60009 |
| [14] | Janson, S. and Kaijser, S. Higher moments of Banach space valued random variables. Mem. Amer. Math. Soc., 238 (1127), vii+110 (2015). MR3402381. · Zbl 0887.60009 |
| [15] | Kallenberg, O. Foundations of modern probability. Probability and its Applications (New York). · Zbl 1336.60007 |
| [16] | Springer-Verlag, New York, second edition (2002). ISBN 0-387-95313-2. MR1876169. · Zbl 0996.60001 |
| [17] | Kanagawa, M., Hennig, P., Sejdinovic, D., and Sriperumbudur, B. K. Gaussian processes and kernel methods: a review on connections and equivalences. ArXiv Mathematics e-prints (2018). arXiv: 1807.02582v1. |
| [18] | Lax, P. D. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York (2002). ISBN 0-471-55604-1. MR1892228. · Zbl 1009.47001 |
| [19] | Lyons, R. Distance covariance in metric spaces. Ann. Probab., 41 (5), 3284-3305 (2013). MR3127883. Errata at Ann. Probab. 46 (2018), no. 4, 2400-2405. MR3813995. · Zbl 1466.62343 |
| [20] | Olver, F. W. J., Lozier, D. W., Boisvert, R. F., and Clark, C. W., editors. NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge (2010). ISBN 978-0-521-14063-8. MR2723248. URL: http://dlmf.nist.gov/. · Zbl 1198.00002 |
| [21] | Pietsch, A. Nukleare lokalkonvexe Räume. Schriftenreihe Inst. Math. Deutsch. Akad. Wiss. Berlin, Reihe A, Reine Mathematik, Heft 1. Akademie-Verlag, Berlin, second edition (1965). MR0181888. English translation in Nuclear locally convex spaces (1972). MR0350360. · Zbl 0152.32302 |
| [22] | Rüschendorf, L. Wasserstein metric (2020). Encyclopedia of Mathematics. Available at https: //www.encyclopediaofmath.org/index.php?title=Wasserstein_metric. |
| [23] | Schoenberg, I. J. Metric spaces and positive definite functions. Trans. Amer. Math. Soc., 44 (3), 522-536 (1938). MR1501980. · Zbl 0019.41502 |
| [24] | Sejdinovic, D., Sriperumbudur, B., Gretton, A., and Fukumizu, K. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Ann. Statist., 41 (5), 2263-2291 (2013). MR3127866. · Zbl 1281.62117 |
| [25] | Székely, G. J. and Rizzo, M. L. Brownian distance covariance. Ann. Appl. Stat., 3 (4), 1236-1265 (2009a). MR2752127. · Zbl 1196.62077 |
| [26] | Székely, G. J. and Rizzo, M. L. Rejoinder: Brownian distance covariance. Ann. Appl. Stat., 3 (4), 1303-1308 (2009b). MR2752135. · Zbl 1196.62077 |
| [27] | Székely, G. J., Rizzo, M. L., and Bakirov, N. K. Measuring and testing dependence by correlation of distances. Ann. Statist., 35 (6), 2769-2794 (2007). MR2382665. · Zbl 1129.62059 |
| [28] | Varadarajan, V. S. On the convergence of sample probability distributions. Sankhyā, 19, 23-26 (1958). MR94839. · Zbl 0082.34201 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
