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THE SUM OF THE RECIPROCAL OF THE RANDOM WALK

Published online by Cambridge University Press:  13 February 2019

Jon Michel  and
Robert de Jong
Jon Michel*
Affiliation:
Ohio State University
Robert de Jong
Affiliation:
Ohio State University
*
*Address correspondence to Jon Michel, Department of Economics, Ohio State University, 304 Arps Hall, Columbus, OH 43210, USA; e-mail: [email protected].
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Abstract

This paper derives the limit distribution of the rescaled sum of the reciprocal of the positive part of a random walk with continuously distributed innovations, and of the rescaled sum of the reciprocal of the absolute value of a random walk with continuously distributed innovations. It also considers this statistic for the case of a simple random walk, and shows that the limit distribution is different for this case.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 1 , February 2020 , pp. 170 - 183
Copyright
Copyright © Cambridge University Press 2019 

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Footnotes

We would like to thank the Editor (Peter C.B. Phillips), Co-Editor (Benedikt Pötscher), and two anonymous referees for numerous useful comments and suggestions.

References

REFERENCES

Akonom, J. (1993) Comportement asymptotique du temps d’occupation du processus des sommes partielles. Annales de l’Institut Henri Poincaré 29, 57–81.Google Scholar
Borodin, A.N. & Ibragimov, I.A. (1995) Limit theorems for functionals of random walks. Proceedings of the Steklov Institute of Mathematics 195(2).Google Scholar
Csáki, E., Csörgő, M., Földes, A., & Révész, P. (1992) Strong approximation of additive functionals. Journal of Theoretical Probability 5, 679–706.CrossRefGoogle 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. (2005) Further results on the asymptotics for nonlinear transformations of integrated time series. Econometric Theory 21, 413–430.CrossRefGoogle Scholar
de Jong, R.M. (2010) Exponentials of Unit Root Processes. Working paper, Ohio State University.Google Scholar
Dobrushin, R.L. (1955) Two limit theorems for the simplest random walk on a line (in Russian). Uspekhi Matematicheskikh Nauk 10(3)(65), 139–146.Google Scholar
Kallianpur, G. & Robbins, H. (1954) The sequence of sums of independent random variables. Duke Mathematical Journal 21(2), 285–307.CrossRefGoogle Scholar
Karatzas, I.& Shreve, S. (1991) Brownian Motion and Stochastic Calculus, 2nd ed. Springer-Verlag.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
Phillips, P.C.B. & Sul, D.. (2007) Transition modeling and econometric convergence tests. Econometrica 75(6), 1771–1855.CrossRefGoogle Scholar
Phillips, P.C.B. & Sul, D.. (2009) Economic transition and growth. Journal of Applied Econometrics 24(7), 1153-1185.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
Qu, X. & de Jong, R.M. (2012) Sums of exponentials of random walks with drift. Econometric Theory 28, 915–924.CrossRefGoogle Scholar
Révész, P. (1981) Local time and invariance. Analytical Methods in Probability Theory, Proceedings, pp. 128–145. Lecture Notes in Mathematics, vol. 861. World Scientific.CrossRefGoogle Scholar
Révész, P. (2005) Random Walk in Random and Non-Random Environments, 2nd ed. World Scientific.CrossRefGoogle Scholar
Revuz, D. & Yor, M. (1999) Continuous Martingales and Brownian Motion, 3rd ed. Springer-Verlag.CrossRefGoogle Scholar