The Annals of Statistics

Asymptotic expansions for sums of block-variables under weak dependence

S. N. Lahiri

Full-text: Open access


Let {Xi}i=−∞ be a sequence of random vectors and $Y_{in}=f_{in}(\mathcal{X}_{i,\ell})$ be zero mean block-variables where $\mathcal{X}_{i,\ell}=(X_{i},\ldots,X_{i+\ell-1})$, i≥1, are overlapping blocks of length and where fin are Borel measurable functions. This paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums ∑i=1nXi and ∑i=1nYin under weak dependence conditions on the sequence {Xi}i=−∞ when the block length grows to infinity. In contrast to the classical Edgeworth expansion results where the terms in the expansions are given by powers of n−1/2, the expansions derived here are mixtures of two series, one in powers of n−1/2 and the other in powers of $[\frac{n}{\ell}]^{-1/2}$. Applications of the main results to (i) expansions for Studentized statistics of time series data and (ii) second order correctness of the blocks of blocks bootstrap method are given.

Article information

Ann. Statist., Volume 35, Number 3 (2007), 1324-1350.

First available in Project Euclid: 26 July 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Blocks of blocks bootstrap Cramér’s condition Edgeworth expansions moderate deviation inequality moving block bootstrap second-order correctness Studentized statistics strong mixing


Lahiri, S. N. Asymptotic expansions for sums of block-variables under weak dependence. Ann. Statist. 35 (2007), no. 3, 1324--1350. doi:10.1214/009053607000000190.

Export citation


  • Babu, G. J. and Singh, K. (1984). On one term Edgeworth correction by Efron's bootstrap. Sankyā Ser. A 46 219–232.
  • Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434–451.
  • Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions. Krieger, Melbourne, FL.
  • Carlstein, E. (1986). The use of subseries methods for estimating the variance of a general statistic from a stationary time series. Ann. Statist. 14 1171–1179.
  • Feller, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • Götze, F. (1987). Approximations for multivariate $U$-statistics. J. Multivariate Anal. 22 212–229.
  • Götze, F. and Hipp, C. (1978). Asymptotic expansions in the central limit theorem under moment conditions. Z. Wahrsch. Verw. Gebiete 42 67–87.
  • Götze, F. and Hipp, C. (1983). Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrsch. Verw. Gebiete 64 211–239.
  • Götze, F. and Hipp, C. (1994). Asymptotic distribution of statistics in time series. Ann. Statist. 22 2062–2088.
  • Götze, F. and Künsch, H. R. (1996). Second-order correctness of the blockwise bootstrap for stationary observations. Ann. Statist. 24 1914–1933.
  • Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • Hall, P., Horowitz, J. L. and Jing, B.-Y. (1995). On blocking rules for the bootstrap with dependent data. Biometrika 82 561–574.
  • Hall, P. and Jing, B.-Y. (1996). On sample reuse methods for dependent data. J. Roy. Statist. Soc. Ser. B 58 727–737.
  • Janas, D. (1994). Edgeworth expansions for spectral mean estimates with applications to Whittle estimates. Ann. Inst. Statist. Math. 46 667–682.
  • Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. Ann. Statist. 25 2084–2102.
  • Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217–1241.
  • Lahiri, S. N. (1993). Refinements in asymptotic expansions for sums of weakly dependent random vectors. Ann. Probab. 21 791–799.
  • Lahiri, S. N. (1996). Asymptotic expansions for sums of random vectors under polynomial mixing rates. Sankhyā Ser. A 58 206–224.
  • Lahiri, S. N. (1996). On Edgeworth expansion and moving block bootstrap for Studentized $M$-estimators in multiple linear regression models. J. Multivariate Anal. 56 42–59.
  • Lahiri, S. N. (1999). Theoretical comparisons of block bootstrap methods. Ann. Statist. 27 386–404.
  • Lahiri, S. N. (2006). Edgeworth expansions for Studentized statistics under weak dependence. Working paper, Dept. Statistics, Iowa State Univ.
  • Lahiri, S. N. (2006). Asymptotic expansions for sums of block-variables under weak dependence. Preprint. Available at
  • Liu, R. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap (R. LePage and L. Billard, eds.) 225–248. Wiley, New York.
  • Politis, D. and Romano, J. P. (1992). A general resampling scheme for triangular arrays of $\alpha$-mixing random variables with application to the problem of spectral density estimation. Ann. Statist. 20 1985–2007.
  • Politis, D. and Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031–2050.
  • Priestley, M. B. (1981). Spectral Analysis and Time Series. 1. Univariate Series. Academic Press, New York.
  • Skovgaard, I. M. (1981). Transformation of an Edgeworth expansion by a sequence of smooth functions. Scand. J. Statist. 8 207–217.
  • Sweeting, T. J. (1977). Speeds of convergence for the multidimensional central limit theorem. Ann. Probab. 5 28–41.
  • Tikhomirov, A. N. (1980). On the convergence rate in the central limit theorem for weakly dependent random variables. Theory Probab. Appl. 25 790–809.
  • Velasco, C. and Robinson, P. M. (2001). Edgeworth expansions for spectral density estimates and Studentized sample mean. Econometric Theory 17 497–539.