The Annals of Statistics

Inference of weighted $V$-statistics for nonstationary time series and its applications

Zhou Zhou

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We investigate the behavior of Fourier transforms for a wide class of nonstationary nonlinear processes. Asymptotic central and noncentral limit theorems are established for a class of nondegenerate and degenerate weighted $V$-statistics through the angle of Fourier analysis. The established theory for $V$-statistics provides a unified treatment for many important time and spectral domain problems in the analysis of nonstationary time series, ranging from nonparametric estimation to the inference of periodograms and spectral densities.

Article information

Ann. Statist., Volume 42, Number 1 (2014), 87-114.

First available in Project Euclid: 15 January 2014

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Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 60F05: Central limit and other weak theorems

$V$-statistics Fourier transform nondegeneracy degeneracy locally stationary time series nonparametric inference spectral analysis


Zhou, Zhou. Inference of weighted $V$-statistics for nonstationary time series and its applications. Ann. Statist. 42 (2014), no. 1, 87--114. doi:10.1214/13-AOS1184.

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  • Beutner, E. and Zähle, H. (2012). Deriving the asymptotic distribution of $U$- and $V$-statistics of dependent data using weighted empirical processes. Bernoulli 18 803–822.
  • Beutner, E. and Zähle, H. (2013). Continuous mapping approach to the asymptotics of $U$- and $V$-statistics. Bernoulli. To appear.
  • Bhansali, R. J., Giraitis, L. and Kokoszka, P. S. (2007). Approximations and limit theory for quadratic forms of linear processes. Stochastic Process. Appl. 117 71–95.
  • Brillinger, D. R. (1969). Asymptotic properties of spectral estimates of second order. Biometrika 56 375–390.
  • Davis, R. A., Lee, T. C. M. and Rodriguez-Yam, G. A. (2006). Structural break estimation for nonstationary time series models. J. Amer. Statist. Assoc. 101 223–239.
  • de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261–277.
  • de Wet, T. and Venter, J. H. (1973). Asymptotic distributions for quadratic forms with applications to tests of fit. Ann. Statist. 1 380–387.
  • Dehling, H. (2006). Limit theorems for dependent $U$-statistics. In Dependence in Probability and Statistics. Lecture Notes in Statist. 187 65–86. Springer, New York.
  • Dehling, H. and Taqqu, M. S. (1989). The empirical process of some long-range dependent sequences with an application to $U$-statistics. Ann. Statist. 17 1767–1783.
  • Dehling, H. and Wendler, M. (2010). Central limit theorem and the bootstrap for $U$-statistics of strongly mixing data. J. Multivariate Anal. 101 126–137.
  • Denker, M. (1985). Asymptotic Distribution Theory in Nonparametric Statistics. Friedr. Vieweg & Sohn, Braunschweig.
  • Dette, H., Preuss, P. and Vetter, M. (2011). A measure of stationarity in locally stationary processes with applications to testing. J. Amer. Statist. Assoc. 106 1113–1124.
  • Dwivedi, Y. and Subba Rao, S. (2011). A test for second order stationarity based on the discrete Fourier transform. J. Time Series Anal. 32 68–91.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. Chapman & Hall, London.
  • Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • Fox, R. and Taqqu, M. S. (1987). Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Related Fields 74 213–240.
  • Gao, J. and Anh, V. (2000). A central limit theorem for a random quadratic form of strictly stationary processes. Statist. Probab. Lett. 49 69–79.
  • Götze, F. and Tikhomirov, A. N. (1999). Asymptotic distribution of quadratic forms. Ann. Probab. 27 1072–1098.
  • Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statistics 19 293–325.
  • Hsing, T. and Wu, W. B. (2004). On weighted $U$-statistics for stationary processes. Ann. Probab. 32 1600–1631.
  • Hušková, M. and Janssen, P. (1993). Consistency of the generalized bootstrap for degenerate $U$-statistics. Ann. Statist. 21 1811–1823.
  • Kuo, H. H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics 463. Springer, Berlin.
  • Lee, A. J. (1990). $U$-statistics: Theory and Practice. Statistics: Textbooks and Monographs 110. Dekker, New York.
  • Lee, J. and Subba Rao, S. (2011). A note on general quadratic forms of nonstationary time series. Preprint.
  • Leucht, A. (2012). Degenerate $U$- and $V$-statistics under weak dependence: Asymptotic theory and bootstrap consistency. Bernoulli 18 552–585.
  • Liflyand, E., Samko, S. and Trigub, R. (2012). The Wiener algebra of absolutely convergent Fourier integrals: An overview. Anal. Math. Phys. 2 1–68.
  • Major, P. (1994). Asymptotic distributions for weighted $U$-statistics. Ann. Probab. 22 1514–1535.
  • O’Neil, K. A. and Redner, R. A. (1993). Asymptotic distributions of weighted $U$-statistics of degree $2$. Ann. Probab. 21 1159–1169.
  • Priestley, M. B. (1981). Spectral Analysis and Time Series. Vol. 1. Academic Press, London.
  • Rifi, M. and Utzet, F. (2000). On the asymptotic behavior of weighted $U$-statistics. J. Theoret. Probab. 13 141–167.
  • Rosenblatt, M. (1984). Asymptotic normality, strong mixing and spectral density estimates. Ann. Probab. 12 1167–1180.
  • Shao, X. and Wu, W. B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773–1801.
  • von Mises, R. (1947). On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statistics 18 309–348.
  • Walker, A. M. (2000). Some results concerning the asymptotic distribution of sample Fourier transforms and periodograms for a discrete-time stationary process with a continuous spectrum. J. Time Series Anal. 21 95–109.
  • Wu, W. B. and Zhou, Z. (2011). Gaussian approximations for non-stationary multiple time series. Statist. Sinica 21 1397–1413.
  • Yoshihara, K.-i. (1976). Limiting behavior of $U$-statistics for stationary, absolutely regular processes. Z. Wahrsch. Verw. Gebiete 35 237–252.
  • Zhou, Z. (2010). Nonparametric inference of quantile curves for nonstationary time series. Ann. Statist. 38 2187–2217.
  • Zhou, Z. (2013). Heteroscedasticity and autocorrelation robust structural change detection. J. Amer. Statist. Assoc. 108 726–740.
  • Zhou, Z. (2014). Supplement to “Inference of weighted $V$-statistics for non-stationary time series and its applications.” DOI:10.1214/13-AOS1184SUPP.
  • Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729.

Supplemental materials

  • Supplementary material: Supplement for “Inference of weighted $V$-statistics for nonstationary time series and its applications”. This supplementary material contains auxiliary lemmas and proofs of Propositions 1, 3, 4 and Corollaries 3, 4 of the paper.