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  • Qiying Wang (a1) and Peter C.B. Phillips (a2)

A local limit theorem is given for the sample mean of a zero energy function of a nonstationary time series involving twin numerical sequences that pass to infinity. The result is applicable in certain nonparametric kernel density estimation and regression problems where the relevant quantities are functions of both sample size and bandwidth. An interesting outcome of the theory in nonparametric regression is that the linear term is eliminated from the asymptotic bias. In consequence and in contrast to the stationary case, the Nadaraya–Watson estimator has the same limit distribution (to the second order including bias) as the local linear nonparametric estimator.

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*Address correspondence to Peter C.B. Phillips, Cowles Foundation, Yale University, P.O. Box 208281, New Haven, CT 06520, USA; e-mail:
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Akonom, J. (1993) Comportement asymptotique du temps d’occupation du processus des sommes partielles. Annals of the Institute of Henri Poincaré 29, 5781.
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Billingsley, P. (1974) Conditional distributions and tightness. Annals of Probability 2, 480485.
Borodin, A.N. & Ibragimov, I.A. (1995) Limit Theorems for Functionals of Random Walks. Proceedings of the Steklov Institute of Mathematics, Vol. 195 (Sudakov, V.N., ed.). American Mathematical Society.
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. Chapman and Hall.
Geman, D. & Horowitz, J. (1980) Occupation densities. Annals of Probability 8, 167.
Hall, P. (1977) Martingale invariance principles. Annals of Probability 5, 875887.
Jeganathan, P. (2004) Convergence of functionals of sums of r.v.s to local times of fractional stable motions. Annals of Probability 32, 17711795.
Jeganathan, P. (2008) Limit Theorems for Functionals of Sums That Converge to Fractional Brownian and Stable Motions. Cowles Foundation Discussion Paper 1949.
Luklacs, E. (1970) Characteristic Functions. Hafner.
Petrov, V.V. (1995) Limit Theorems of Probability Theory. Oxford University Press.
Phillips, P.C.B. & Park, J.Y. (1998) Nonstationary Density Estimation and Kernel Autoregression. Cowles Foundation Discussion Paper 1181.
Wang, Q., Lin, Y.-X., & Gulati, C.M. (2003) Asymptotics for general fractionally integrated processes with applications to unit root tests. Econometric Theory 19, 143164.
Wang, Q. & Phillips, P.C.B. (2009a) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710738.
Wang, Q. & Phillips, P.C.B. (2009b) Structural Nonparametric Cointegrating Regression. Econometrica 77, 19011948.
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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