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A more general central limit theorem for m-dependent random variables with unbounded m


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  • Romano, Joseph P.
  • Wolf, Michael


In this article, a general central limit theorem for a triangular array of m-dependent random variables is presented. Here, m may tend to infinity with the row index at a certain rate. Our theorem is a generalization of previous results. Some examples are given that show that the generalization is useful. In particular, we consider the limiting behavior of the sample mean of a combined sample of independent long-memory sequences, the limiting behavior of a spectral estimator, and the moving blocks bootstrap distribution. The examples make it clear the consideration of asymptotic behavior with the amount of dependence m increasing with n is useful even when the underlying processes are weakly dependent (or even independent), because certain natural statistics that arise in the analysis of time series have this structure. In addition, we provide an example to demonstrate the sharpness of our result.

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Bibliographic Info

Article provided by Elsevier in its journal Statistics & Probability Letters.

Volume (Year): 47 (2000)
Issue (Month): 2 (April)
Pages: 115-124

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Handle: RePEc:eee:stapro:v:47:y:2000:i:2:p:115-124

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Keywords: Central limit theorem m-dependent random variables;


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Cited by:
  1. Le Breton, Michel & Lepelley, Dominique & Smaoui, Hatem, 2012. "The Probability of Casting a Decisive Vote: From IC to IAC trhough Ehrhart's Polynomials and Strong Mixing," IDEI Working Papers 722, Institut d'Économie Industrielle (IDEI), Toulouse.
  2. Jacek Leśkow & Rafał Synowiecki, 2010. "On bootstrapping periodic random arrays with increasing period," Metrika, Springer, vol. 71(3), pages 253-279, May.
  3. Zhao, Zhibiao, 2010. "Density estimation for nonlinear parametric models with conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 155(1), pages 71-82, March.
  4. Geenens, Gery & Simar, Léopold, 2010. "Nonparametric tests for conditional independence in two-way contingency tables," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 765-788, April.
  5. Zhao, Zhibiao & Wu, Wei Biao, 2007. "Asymptotic theory for curve-crossing analysis," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 862-877, July.
  6. Harvey, Danielle J. & Weng, Qian & Beckett, Laurel A., 2010. "On an asymptotic distribution of dependent random variables on a 3-dimensional lattice," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1015-1021, June.


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