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CENTRAL LIMIT THEOREMS FOR WEIGHTED SUMS OFLINEAR PROCESSES: LP-APPROXIMABILITY VERSUS BROWNIANMOTION

Published online by Cambridge University Press:  01 June 2009

Kairat T. Mynbaev
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Abstract

Standardized slowly varying regressors are shown to beLp-approximable. Thisfact allows us to provide alternative proofs ofasymptotic expansions of nonstochastic quantitiesand central limit results due to P.C.B. Phillips,under a less stringent assumption on linearprocesses. The recourse to stochastic calculusrelated to Brownian motion can be completelydispensed with.

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Type
Research Article
Information
Econometric Theory , Volume 25 , Issue 3 , June 2009 , pp. 748 - 763
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

The paper has been significantly improvedfollowing comments and suggestions by threeanonymous referees and an associate editor. I amespecially grateful to one of the referees for acryptic remark that my paper “opens the way to usethe important results of Cremers and Kadelka(1986).”

References

REFERENCES

Aljančić, S., Bojanić, R., & Tomić, M. (1955) Deux théorèmes relatifs au comportement asymptotique des séries trigonométriques. Zbornik Radova Matematički Institut SANU 43, 15–26.Google Scholar
Anderson, T.W. (1971) The Statistical Analysis of Time Series. Wiley.Google Scholar
Beveridge, S. & Nelson, C.R. (1981) A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the “business cycle.” Journal of Monetary Economics 7, 151–174.10.1016/0304-3932(81)90040-4CrossRefGoogle Scholar
Cremers, H. & Kadelka, D. (1986) On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in . Stochastic Processes and Their Applications 21, 305–317.10.1016/0304-4149(86)90102-XCrossRefGoogle Scholar
de Haan, L. & Resnick, S. (1996) Second-order regular variation and rates of convergence in extreme-value theory. Annals of Probability 24, 97–124.Google Scholar
de Jong, R.M. & Davidson, J. (2000) The functional central limit theorem and weak convergence to stochastic integrals part, I: Weakly dependent processes. Econometric Theory 16, 621–642.10.1017/S0266466600165016CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley.10.1002/9780470316429CrossRefGoogle Scholar
Hannan, E.J. (1979) The central limit theorem for time series regression. Stochastic Processes and Their Applications 9, 281–289.10.1016/0304-4149(79)90050-4CrossRefGoogle Scholar
Moussatat, M.W. (1976) On the asymptotic theory of statistical experiments and some of its applications. Ph.D. Dissertation, University of California, Berkeley.Google Scholar
Mynbaev, K.T. (2001) Lp-approximable sequences of vectors and limit distribution of quadratic forms of random variables. Advances in Applied Mathematics 26, 302–329.10.1006/aama.2001.0723CrossRefGoogle Scholar
Mynbaev, K.T. & Ullah, A. (2008) Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model. Journal of Multivariate Analysis 99, 245–277.10.1016/j.jmva.2007.04.002CrossRefGoogle Scholar
Phillips, P.C.B. (2007) Regression with slowly varying regressors and nonlinear trends. Econometric Theory 23, 557–614.10.1017/S0266466607070260CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.10.1214/aos/1176348666CrossRefGoogle Scholar
Pötscher, B.M. & Prucha, I.R. (1991) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part I: Consistency and approximation concepts. Econometric Reviews 10, 125–216. Available at http://gemini.econ.umd.edu/jrust/econ551/syb551/node5.html.10.1080/07474939108800204CrossRefGoogle Scholar
Seneta, E. (1985) Pravil′no menyayushchiesya funktsii (in Russian) [Regularly varying functions] with appendixes by I.S. Shiganov and V.M. Zolotarev. “Nauka,” Moscow.Google Scholar
Tanaka, K. (1996) Time Series Analysis. Wiley.Google Scholar