The Annals of Statistics

Self-normalized Cramér-type moderate deviations under dependence

Xiaohong Chen, Qi-Man Shao, Wei Biao Wu, and Lihu Xu

Full-text: Open access


We establish a Cramér-type moderate deviation result for self-normalized sums of weakly dependent random variables, where the moment requirement is much weaker than the non-self-normalized counterpart. The range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes. We consider three types of self-normalization: the equal-block scheme, the big-block-small-block scheme and the interlacing scheme. Simulation study shows that the latter can have a better finite-sample performance. Our result is applied to multiple testing and construction of simultaneous confidence intervals for ultra-high dimensional time series mean vectors.

Article information

Ann. Statist., Volume 44, Number 4 (2016), 1593-1617.

Received: April 2015
Revised: December 2015
First available in Project Euclid: 7 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 60F10: Large deviations

Cramér-type moderate deviation absolutely regular functional dependence measures ultra-high dimensional time series


Chen, Xiaohong; Shao, Qi-Man; Wu, Wei Biao; Xu, Lihu. Self-normalized Cramér-type moderate deviations under dependence. Ann. Statist. 44 (2016), no. 4, 1593--1617. doi:10.1214/15-AOS1429.

Export citation


  • [1] Andrews, D. W. K. (1984). Nonstrong mixing autoregressive processes. J. Appl. Probab. 21 930–934.
  • [2] Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95–115.
  • [3] Beare, B. K. (2010). Copulas and temporal dependence. Econometrica 78 395–410.
  • [4] Berbee, H. (1987). Convergence rates in the strong law for bounded mixing sequences. Probab. Theory Related Fields 74 255–270.
  • [5] Bercu, B. and Touati, A. (2008). Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18 1848–1869.
  • [6] Bradley, R. (2007). Introduction to Strong Mixing Conditions. Kendrick Press, Heber City, UT.
  • [7] Bühlmann, P. (2002). Bootstraps for time series. Statist. Sci. 17 52–72.
  • [8] Burkholder, D. L. (1988). Sharp inequalities for martingales and stochastic integrals. Astérisque 157–158 75–94.
  • [9] Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18 17–39.
  • [10] Chang, J., Chen, S. X. and Chen, X. (2015). High dimensional generalized empirical likelihood for moment restrictions with dependent data. J. Econometrics 185 283–304.
  • [11] Chen, X., Shao, Q., Wu, W. and Xu, L. (2016). Supplement to “Self-normalized Cramér-type moderate deviations under dependence.” DOI:10.1214/15-AOS1429SUPP.
  • [12] Chen, R. and Tsay, R. (1993). Nonlinear additive ARX models. J. Amer. Statist. Assoc. 88 955–967.
  • [13] Chen, R. and Tsay, R. S. (1993). Functional-coefficient autoregressive models. J. Amer. Statist. Assoc. 88 298–308.
  • [14] Chen, S. X. and Qin, Y. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Statist. 38 808–835.
  • [15] Chen, X. (2012). Penalized sieve estimation and inference of semi-nonparametric dynamic models: A selective review. In Advances in Economics and Econometrics, 2010 World Congress of the Econometric Society Book Volumes Cambridge Univ. Press, Cambridge.
  • [16] Chen, X., Hansen, L. P. and Carrasco, M. (2010). Nonlinearity and temporal dependence. J. Econometrics 155 155–169.
  • [17] Chen, X., Wu, W. B. and Yi, Y. (2009). Efficient estimation of copula-based semiparametric Markov models. Ann. Statist. 37 4214–4253.
  • [18] Davydov, Y. A. (1968). Convergence of distributions generated by stationary stochastic processes. Theory Probab. Appl. 13 691–696.
  • [19] Dedecker, J., Doukhan, P., Lang, G., León, J. R., Louhichi, S. and Prieur, C. (2007). Weak Dependence: With Examples and Applications. Lecture Notes in Statistics 190. Springer, New York.
  • [20] Delaigle, A., Hall, P. and Jin, J. (2011). Robustness and accuracy of methods for high dimensional data analysis based on Student’s $t$-statistic. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 283–301.
  • [21] de la Peña, V. H., Lai, T. L. and Shao, Q. (2009). Self-Normalized Processes: Limit Theory and Statistical Applications. Springer, Berlin.
  • [22] Douc, R., Moulines, E., Olsson, J. and van Handel, R. (2011). Consistency of the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 39 474–513.
  • [23] Doukhan, P. and Wintenberger, O. (2008). Weakly dependent chains with infinite memory. Stochastic Process. Appl. 118 1997–2013.
  • [24] Fan, J., Hall, P. and Yao, Q. (2007). To how many simultaneous hypothesis tests can normal, Student’s $t$ or bootstrap calibration be applied? J. Amer. Statist. Assoc. 102 1282–1288.
  • [25] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • [26] Jing, B., Shao, Q. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167–2215.
  • [27] Juodis, M. and Račkauskas, A. (2005). A remark on self-normalization for dependent random variables. Lith. Math. J. 45 142–151.
  • [28] Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer, New York.
  • [29] Lange, T. (2011). Tail behavior and OLS estimation in AR–GARCH models. Statist. Sinica 21 1191–1200.
  • [30] Lin, D. and Foster, D. (2014). The power of a few large blocks: A credible assumption with incredible efficiency. Working paper.
  • [31] Liu, W. and Shao, Q. (2013). A Carmér moderate deviation theorem for Hotelling’s $T^{2}$-statistic with applications to global tests. Ann. Statist. 41 296–322.
  • [32] Liu, W., Shao, Q. and Wang, Q. (2013). Self-normalized Cramér type moderate deviations for the maximum of sums. Bernoulli 19 1006–1027.
  • [33] Masry, E. and Tjøstheim, D. (1995). Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory 11 258–289.
  • [34] Meitz, M. and Saikkonen, P. (2011). Parameter estimation in nonlinear AR–GARCH models. Econometric Theory 27 1236–1278.
  • [35] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • [36] Pham, T. D. and Tran, L. T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297–303.
  • [37] Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
  • [38] Sakhanenko, A. I. (1991). Estimates of Berry–Esseen type for the probabilities of large deviations. Siberian Math. J. 32 647–656.
  • [39] Shao, Q. and Wang, Q. (2013). Self-normalized limit theorems: A survey. Probab. Surv. 10 69–93.
  • [40] Shao, Q. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098–2127.
  • [41] Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. Clarendon Press, Oxford.
  • [42] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
  • [43] Wang, Q. and Hall, P. (2009). Relative errors in central limit theorems for Student’s $t$ statistic, with applications. Statist. Sinica 19 343–354.
  • [44] Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154 (electronic).
  • [45] Wu, W. B. (2011). Asymptotic theory for stationary processes. Stat. Interface 4 207–226.
  • [46] Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436.

Supplemental materials