×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Arcones, M.A. and Giné, E. (1992). On the bootstrap of \(U\) and \(V\) statistics. Ann. Statist. 20 655-674. · Zbl 0760.62018
[2] Bickel, P.J. and Freedman, D.A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196-1217. · Zbl 0449.62034
[3] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley. · Zbl 0172.21201
[4] Davis, R.A., Matsui, M., Mikosch, T. and Wan, P. (2018). Applications of distance correlation to time series. Bernoulli 24 3087-3116. · Zbl 1414.62357
[5] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer. · Zbl 1101.62002
[6] de Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in \(C[0,1]\). Ann. Probab. 29 467-483. · Zbl 1010.62016
[7] Dehling, H., Matsui, M., Mikosch, T., Samorodnitsky, G. and Tafakori, L. (2020). Supplement to “Distance covariance for discretized stochastic processes.” https://doi.org/10.3150/20-BEJ1206SUPP
[8] Dehling, H. and Mikosch, T. (1994). Random quadratic forms and the bootstrap for \(U\)-statistics. J. Multivariate Anal. 51 392-413. · Zbl 0815.62028
[9] Feuerverger, A. (1993). A consistent test for bivariate dependence. Int. Stat. Rev. 61 419-433. · Zbl 0826.62032
[10] Hoffmann-Jørgensen, J. (1994). Probability with a View Toward Statistics, Vol. I. Chapman & Hall Probability Series. New York: CRC Press. · Zbl 0821.62003
[11] Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer Series in Statistics. New York: Springer. · Zbl 1279.62017
[12] Lyons, R. (2013). Distance covariance in metric spaces. Ann. Probab. 41 3284-3305. · Zbl 1292.62087
[13] Lyons, R. (2018). Errata to “Distance covariance in metric spaces” [MR3127883]. Ann. Probab. 46 2400-2405. · Zbl 1466.62343
[14] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12 1099-1126. · Zbl 1126.60038
[15] Matsui, M., Mikosch, T. and Samorodnitsky, G. (2017). Distance covariance for stochastic processes. Probab. Math. Statist. 37 355-372. · Zbl 1396.62115
[16] Samorodnitsky, G. (2016). Stochastic Processes and Long Range Dependence. Springer Series in Operations Research and Financial Engineering. Cham: Springer. · Zbl 1376.60007
[17] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling. New York: CRC Press. · Zbl 0925.60027
[18] Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. · Zbl 0538.62002
[19] Székely, G.J. and Rizzo, M.L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236-1265. · Zbl 1196.62077
[20] Székely, G.J. and Rizzo, M.L. (2013). The distance correlation \(t\)-test of independence in high dimension. J. Multivariate Anal. 117 193-213. · Zbl 1277.62128
[21] Székely, G.J. and Rizzo, M.L. (2014). Partial distance correlation with methods for dissimilarities. Ann. Statist. 42 2382-2412. · Zbl 1309.62105
[22] Székely, G. · Zbl 1129.62059
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.