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WEAK CONVERGENCE OF NONLINEAR TRANSFORMATIONS OF INTEGRATED PROCESSES: THE MULTIVARIATE CASE

Published online by Cambridge University Press:  01 October 2009

Norbert Christopeit
Norbert Christopeit*
Affiliation:
University of Bonn
*
*Address correspondence to Norbert Christopeit, Institute of Econometrics, Department of Economics, University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany; e-mail: [email protected].
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Abstract

We consider weak convergence of sample averages of nonlinearly transformed stochastic triangular arrays satisfying a functional invariance principle. A fundamental paradigm for such processes is constituted by integrated processes. The results obtained are extensions of recent work in the literature to the multivariate and non-Gaussian case. As admissible nonlinear transformation, a new class of functionals (so-called locally p-integrable functions) is introduced that adapts the concept of locally integrable functions in Pötscher (2004, Econometric Theory 20, 1–22) to the multidimensional setting.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 5 , October 2009 , pp. 1180 - 1207
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Berkes, I. & Horváth, L. (2006) Convergence of integral functionals of stochastic processes. Econometric Theory 22, 304–322.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Christopeit, N. (1983) Discrete approximation of continuous time stochastic control systems. SIAM Journal on Control and Optimization 21, 17–40.Google Scholar
De Jong, R. (2004) Addendum to “Asymptotics for nonlinear transformation of integrated time series.” Econometric Theory 21, 623–635.Google Scholar
Friedman, A. (1964) Partial Differential Equations of Parbolic Type. Prentice-Hall.Google Scholar
Friedman, A. (1975) Stochastic Differential Equations and Applications, vol. 1. Prentice-Hall.Google Scholar
Hewitt, E. & Stromberg, K. (1969) Real and Abstract Analysis. Springer-Verlag.Google Scholar
Johansen, S. (1995) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press.Google Scholar
Karatzas, I. & Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus, 2nd ed.Springer-Verlag.Google Scholar
Kawata, T. (1972) Fourier Analysis in Probability Theory. Academic Press.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269–298.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regression with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Pötscher, B. (2004) Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem. Econometric Theory 20, 1–22.CrossRefGoogle Scholar
Shiryaev, A.N. (1996) Probability. Springer-Verlag.CrossRefGoogle Scholar
Stroock, D.W. & Varadhan, S.R.S. (1979) Multidimensional Diffusion Processes. Springer-Verlag.Google Scholar