• Bernoulli
  • Volume 24, Number 4A (2018), 3087-3116.

Applications of distance correlation to time series

Richard A. Davis, Muneya Matsui, Thomas Mikosch, and Phyllis Wan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The use of empirical characteristic functions for inference problems, including estimation in some special parametric settings and testing for goodness of fit, has a long history dating back to the 70s. More recently, there has been renewed interest in using empirical characteristic functions in other inference settings. The distance covariance and correlation, developed by Székely et al. (Ann. Statist. 35 (2007) 2769–2794) and Székely and Rizzo (Ann. Appl. Stat. 3 (2009) 1236–1265) for measuring dependence and testing independence between two random vectors, are perhaps the best known illustrations of this. We apply these ideas to stationary univariate and multivariate time series to measure lagged auto- and cross-dependence in a time series. Assuming strong mixing, we establish the relevant asymptotic theory for the sample auto- and cross-distance correlation functions. We also apply the auto-distance correlation function (ADCF) to the residuals of an autoregressive processes as a test of goodness of fit. Under the null that an autoregressive model is true, the limit distribution of the empirical ADCF can differ markedly from the corresponding one based on an i.i.d. sequence. We illustrate the use of the empirical auto- and cross-distance correlation functions for testing dependence and cross-dependence of time series in a variety of contexts.

Article information

Bernoulli, Volume 24, Number 4A (2018), 3087-3116.

Received: July 2016
Revised: February 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

$U$-statistics AR process auto- and cross-distance correlation function ergodicity Fourier analysis residuals strong mixing testing independence time series


Davis, Richard A.; Matsui, Muneya; Mikosch, Thomas; Wan, Phyllis. Applications of distance correlation to time series. Bernoulli 24 (2018), no. 4A, 3087--3116. doi:10.3150/17-BEJ955.

Export citation


  • [1] Aaronson, J., Burton, R., Dehling, H., Gilat, D., Hill, T. and Weiss, B. (1996). Strong laws for $L$- and $U$-statistics. Trans. Amer. Math. Soc. 348 2845–2866.
  • [2] Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42 1656–1670.
  • [3] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [4] Csörgő, S. (1981). Limit behaviour of the empirical characteristic function. Ann. Probab. 9 130–144.
  • [5] Csörgő, S. (1981). Multivariate characteristic functions and tail behaviour. Z. Wahrsch. Verw. Gebiete 55 197–202.
  • [6] Csörgő, S. (1981). Multivariate empirical characteristic functions. Z. Wahrsch. Verw. Gebiete 55 203–229.
  • [7] Davis, R.A., Matsui, M., Mikosch, T. and Wan, P. (2017). Supplement to “Applications of distance correlation to time series.” DOI:10.3150/17-BEJ955SUPP.
  • [8] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. New York: Springer.
  • [9] Dueck, J., Edelmann, D., Gneiting, T. and Richards, D. (2014). The affinely invariant distance correlation. Bernoulli 20 2305–2330.
  • [10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. New York: Wiley.
  • [11] Feuerverger, A. (1993). A consistent test for bivariate dependence. Int. Stat. Rev. 61 419–433.
  • [12] Feuerverger, A. and Mureika, R.A. (1977). The empirical characteristic function and its applications. Ann. Statist. 5 88–97.
  • [13] Fokianos, K. and Pitsillou, M. (2017). Consistent testing for pairwise dependence in time series. Technometrics 59 262–270.
  • [14] Haslett, J. and Raftery, A.E. (1989). Space-time modelling with long-memory dependence: Assessing Ireland’s wind power resource. J. R. Stat. Soc. Ser. C. Appl. Stat. 38 1–50.
  • [15] Hlávka, Z., Hušková, M. and Meintanis, S.G. (2011). Tests for independence in non-parametric heteroscedastic regression models. J. Multivariate Anal. 102 816–827.
  • [16] Hong, Y. (1999). Hypothesis testing in time series via the empirical characteristic function: A generalized spectral density approach. J. Amer. Statist. Assoc. 94 1201–1220.
  • [17] Ibragimov, I.A. and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff Publishing.
  • [18] Krengel, U. (1985). Ergodic Theorems. De Gruyter Studies in Mathematics 6. Berlin: de Gruyter.
  • [19] Kuo, H.H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics 463. Berlin: Springer.
  • [20] Lyons, R. (2013). Distance covariance in metric spaces. Ann. Probab. 41 3284–3305.
  • [21] Meintanis, S.G. and Iliopoulos, G. (2008). Fourier methods for testing multivariate independence. Comput. Statist. Data Anal. 52 1884–1895.
  • [22] Meintanis, S.G., Ngatchou-Wandji, J. and Taufer, E. (2015). Goodness-of-fit tests for multivariate stable distributions based on the empirical characteristic function. J. Multivariate Anal. 140 171–192.
  • [23] Politis, D.N., Romano, J.P. and Wolf, M. (1999). Subsampling. Springer Series in Statistics. New York: Springer.
  • [24] Rémillard, B. (2009). Discussion of: Brownian distance covariance [MR2752127]. Ann. Appl. Stat. 3 1295–1298.
  • [25] Samorodnitsky, G. (2016). Stochastic Processes and Long Range Dependence. Springer Series in Operations Research and Financial Engineering. Cham: Springer.
  • [26] Sejdinovic, D., Sriperumbudur, B., Gretton, A. and Fukumizu, K. (2013). Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Ann. Statist. 41 2263–2291.
  • [27] Székely, G.J. and Rizzo, M.L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236–1265.
  • [28] Székely, G.J. and Rizzo, M.L. (2014). Partial distance correlation with methods for dissimilarities. Ann. Statist. 42 2382–2412.
  • [29] Székely, G.J., Rizzo, M.L. and Bakirov, N.K. (2007). Measuring and testing dependence by correlation of distances. Ann. Statist. 35 2769–2794.
  • [30] Zhou, Z. (2012). Measuring nonlinear dependence in time-series, a distance correlation approach. J. Time Series Anal. 33 438–457.

Supplemental materials

  • Supplement to “Applications of distance correlation to time series”. Complementary results to the proofs of Theorem 4.2 and Lemma 4.1 are provided in the supplement.