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NEGATIVE POWERS OF INTEGRATED PROCESSES

Published online by Cambridge University Press:  30 April 2021

Neslihan Sakarya  and
Robert M. de Jong
Neslihan Sakarya*
Affiliation:
University of Essex
Robert M. de Jong
Affiliation:
Ohio State University
*
Author correspondence to Neslihan Sakarya, Department of Economics, University of Essex, Colchester, UK; e-mail: ns17003@essex.ac.uk.
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Abstract

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This paper derives the limit distribution of the rescaled sum of the absolute value of an integrated process with continuously distributed innovations raised to a negative power less than $-$1, and of the analogous statistic that is obtained using the same function of an integrated process but only considering positive values of the integrated process. We show that the limit behavior of this statistic is determined by the values of the integrated process that are closest to 0, and find the limit behavior of the values of the integrated process that are closest to 0.

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 2 , April 2022 , pp. 339 - 369
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Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

REFERENCES

Akonom, J. (1993) Comportement asymptotique du temps d’occupation du processus des sommes partielles. Annales de l’I.H.P. Probabilités et Statistiques 29, 57–81.Google Scholar
Berkes, I. & Horváth, L. (2006) Convergence of integral functionals of stochastic processes. Econometric Theory 22, 304–322.10.1017/S0266466606060130CrossRefGoogle Scholar
Borodin, A.N. & Ibragimov, I.A. (1995) Limit theorems for functionals of random walks. In Proceedings of the Steklov Institute of Mathematics, vol. 195. American Mathematical Society.Google Scholar
Christopeit, N. (2009) Weak convergence of nonlinear transformations of integrated processes: the multivariate case. Econometric Theory 25, 1180–1207.10.1017/S0266466608090476CrossRefGoogle Scholar
Chung, K.L. (2001) A Course in Probability Theory, 3rd Edition. Academic Press.Google Scholar
de Jong, R.M. (2004) Addendum to “Asymptotics for nonlinear transformations of integrated time series”. Econometric Theory 20, 627–635.CrossRefGoogle Scholar
de Jong, R.M. & Wang, C.H. (2005) Further results on the asymptotics for nonlinear transformations of integrated time series. Econometric Theory 21, 413–430.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, 2nd Edition. vol. II. John Wiley & Sons, Inc.Google Scholar
Fréchet, M. & Shohat, J. (1931) A proof of the generalized second-limit theorem in the theory of probability. Transactions of the American Mathematical Society 33, 533–543.10.1090/S0002-9947-1931-1501604-6CrossRefGoogle Scholar
Jeganathan, P. (2004) Convergence of functionals of sums of r.v.s to local times of fractional stable motions. The Annals of Probability 32, 1771–1795.CrossRefGoogle Scholar
Lin, G.D. (2017) Recent developments on the moment problem. Journal of Statistical Distributions and Applications 4, 1–17.CrossRefGoogle Scholar
Michel, J. & de Jong, R.M. (2020) The sum of the reciprocal of the random walk. Econometric Theory 36, 170–183.CrossRefGoogle Scholar
Miller, K.S. & Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley.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
Pötscher, B.M. (2004) Nonlinear functions and convergence to Brownian motion: beyond the continuous mapping theorem. Econometric Theory 20, 1–22.CrossRefGoogle Scholar
Pötscher, B.M. (2013) On the order of magnitude of sums of negative powers of integrated processes. Econometric Theory 29, 642–658.CrossRefGoogle Scholar
Van der Vaart, A.W. (2000) Asymptotic Statistics. Cambridge University Press.Google Scholar