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On the Structure and Estimation of Reflection Positive Processes

Part of: Inference from stochastic processes Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

R. McVinish
R. McVinish*
Affiliation:
Queensland University of Technology
*
∗Postal address: School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4001, Australia. Email address: r.mcvinish@qut.edu.au
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Abstract

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The class of processes formed as the aggregation of Ornstein-Uhlenbeck processes has proved useful in modeling time series from a number of areas and includes several interesting special cases. This paper examines the second-order properties of this class. Bounds on the one-step prediction error variance are proved and consistency of the minimum contrast estimation is demonstrated.

Keywords

Aggregationlong memoryminimum contrast estimationprediction error

MSC classification

Secondary: 62M15: Spectral analysis 62G05: Estimation

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 150 - 162
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, New York.Google Scholar
[2] Angulo, J. M., Anh, V. V., McVinish, R. and Ruiz-Medina, M. D. (2005). Fractional kinetic equations driven by Gaussian and infinitely divisible noise. Adv. Appl. Prob. 37, 366–392.Google Scholar
[3] Anh, V. V. and McVinish, R. (2003). Completely monotone property of fractional Green functions. Frac. Calc. Appl. Anal. 6, 157–173.Google Scholar
[4] Anh, V. V., Heyde, C. C. and Leonenko, N. N. (2002). Dynamic models of long memory processes driven by Lévy processes. J. Appl. Prob. 39, 730–747.Google Scholar
[5] Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.CrossRefGoogle Scholar
[6] Barndorff-Neilsen, O. E. and Leonenko, N. N. (2005). Spectral properties of superpositions of Ornstein–Uhlenbeck type processes. Methodology Comput. Appl. Prob. 7, 335–352.Google Scholar
[7] Barndorff-Neilsen, O. E. and Pérez-Abreu, V. (1999). Stationary and self-similar processes driven by Lévy processes. Stoch. Process. Appl. 84, 357–369.CrossRefGoogle Scholar
[8] Barndorff-Neilsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck based models and some of their uses in financial econometrics (with discussion). J. R. Statist. Soc. Ser. B 63, 167–241.Google Scholar
[9] Barndorff-Nielsen, O. E., Jensen, J. L. and Sørensen, M. (1998). Some stationary processes in discrete and continuous time. Adv. Appl. Prob. 30, 989–1007.Google Scholar
[10] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
[11] Brillinger, D. R. (1981). Time Series: Data Analysis and Theory. Holden-Day, San Francisco, CA.Google Scholar
[12] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. Springer, New York.Google Scholar
[13] Burg, J. P. (1975). Maximum Entropy and Spectral Analysis. , Stanford Univeristy.Google Scholar
[14] Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein–Uhlenbeck processes. Electron. J. Prob. 8, 1–14.Google Scholar
[15] Comte, F. and Renault, E. (1996). Long memory continuous-time models. J. Econometrics 73, 101–149.CrossRefGoogle Scholar
[16] Cox, D. R. (1991). Long-range dependence, non-linearity and time irreversibility. J. Time Ser. Anal. 12, 329–335.Google Scholar
[17] Dacunha-Castelle, D. and Fermı´n, L. (2006). Disaggregation of long memory processes on C{∞} class. Electron. Commun. Prob. 11, 35–44.Google Scholar
[18] Djrbashian, M. M. (1993). Harmonic Analysis and Boundary Value Problems in the Complex Domain. Birkäuser, Basel.CrossRefGoogle Scholar
[19] Feller, W. (1939). Interpolation of completely monotone functions. Duke Math. J. 5, 661–674.Google Scholar
[20] Gao, J. (2004). Modelling long-range dependent Gaussian processes with application in continuous-time financial models. J. Appl. Prob. 41, 467–482.Google Scholar
[21] Gneiting, T. (2000). Power-law correlations, related models for long-range dependence and their simulation. J. Appl. Prob. 37, 1104–1109.Google Scholar
[22] Granger, C. W. J. (1980). Long memory relationships and the aggregation of dynamic models. J. Econometrics 14, 227–238.Google Scholar
[23] Granger, C. W. J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Series Anal. 1, 15–29.Google Scholar
[24] Hida, T. and Steit, L. (1977). On quantum theory in terms of white noise. Nagoya Math. J. 68, 21–34.CrossRefGoogle Scholar
[25] Igloi, E. and Terdik, G. (1999). Long-range dependence through Gamma-mixed Ornstein–Uhlenbeck processes. Electron. J. Prob. 4, 1–33.Google Scholar
[26] Inoue, A. (1993). On the equations of stationary processes with divergent diffusion coefficients. J. Fac. Sci. Univ. Tokyo Sect. 1A 40, 307–336.Google Scholar
[27] Karlin, S. and Shapley, L. S. (1953). Geometry of Moment Spaces (Amer. Math. Soc. Memoirs 12). American Mathematical Society, Providence, RI.Google Scholar
[28] Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality (Appl. Math. Sci. 28). Springer, New York.CrossRefGoogle Scholar
[29] Leipus, R., Oppenheim, G., Philippe, A. and Viano, M.-C. (2006). Orthogonal series density estimation in a disaggregation scheme. J. Statist. Planning Infer. 136, 2547–2571.Google Scholar
[30] Lin, G. D. (1998). On the Mittag–Leffler distributions. J. Statist. Planning Infer. 74, 1–9.Google Scholar
[31] Martin, R. J. and Walker, A. M. (1997). A power-law model and other models for long-range dependence. J. Appl. Prob. 34, 657–670.Google Scholar
[32] Okabe, Y. (1986). On KMO-Langevin equations for stationary Gaussian process with T-positivity. J. Fac. Sci. Univ. Tokyo Sect. 1A 33, 1–56.Google Scholar
[33] Ostervalder, K. and Schrader, R. (1973). Axioms for Euclidean Green functions. Commun. Math. Phys. 31, 83–112.Google Scholar
[34] Prudikov, A., Brychkov, Y. and Marichev, O. (1990). Integrals and Series, Vol. 5. Gordon and Breach, New York.Google Scholar
[35] Ranga Rao, R. (1962). Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 32, 659–680.Google Scholar
[36] Taqqu, M. S., Willinger, W. and Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev. 27, 5–23.Google Scholar
[37] Viano, M.-C., Deniau, C. and Oppenheim, G. (1995). Long-range dependence and mixing for discrete time fractional processes. J. Time Series Anal. 16, 323–338.Google Scholar
[38] Widder, D. V. (1941). The Laplace Transform. Princeton University Press.Google Scholar
[39] Williams, D. (1991). Probability with Martingales. Cambridge University Press.Google Scholar
[40] Zaffaroni, P. (2004). Contemporaneous aggregation of linear dynamic models in large economies. J. Econometrics 120, 75–82.Google Scholar