Hostname: page-component-89b8bd64d-b5k59 Total loading time: 0 Render date: 2026-05-07T12:28:50.646Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation with ergodic processes and testability

Part of: Nonparametric inference Ergodic theory

Published online by Cambridge University Press:  23 January 2024

Isaac Loh Open the ORCID record for Isaac Loh [Opens in a new window]
Isaac Loh*
Affiliation:
UNC Wilmington
*
*Postal address: Department of Economics and Finance, UNC Wilmington, 601 South College Road, Wilmington NC 28403. Email: lohi@uncw.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that stationary time series can be uniformly approximated over all finite time intervals by mixing, non-ergodic, non-mean-ergodic, and periodic processes, and by codings of aperiodic processes. A corollary is that the ergodic hypothesis—that time averages will converge to their statistical counterparts—and several adjacent hypotheses are not testable in the non-parametric case. Further Baire category implications are also explored.

Keywords

Mixingtestabilityergodicity

MSC classification

Primary: 37A50: Relations with probability theory and stochastic processes
Secondary: 62G10: Hypothesis testing

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 3 , September 2024 , pp. 1040 - 1059
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Adams, T. M. and Nobel, A. B. (1998). On density estimation from ergodic processes. Ann. Prob. 26, 794–804.Google Scholar
Ageev, O. N. and Silva, C. E. (2002). Genericity of rigid and multiply recurrent infinite measure-preserving and nonsingular transformations. Topology Proc. 26, 357–365.Google Scholar
Alpern, S. and Prasad, V. S. (1989). Coding a stationary process to one with prescribed marginals. Ann. Prob. 17, 1658–1663.Google Scholar
Alpern, S. and Prasad, V. S. (2008). Multitowers, conjugacies and codes: Three theorems in ergodic theory, one variation on Rokhlin’s lemma. Proc. Am. Math. Soc. 136, 4373–4383.Google Scholar
Bezuglyi, S., Kwiatkowski, J., and Medynets, K. (2005). Approximation in ergodic theory, Borel and Cantor dynamics. Contemp. Math. 385, 39–64.Google Scholar
Billingsley, P. (1995). Probability and Measure. Wiley, Chichester.Google Scholar
Bogachev, V. I. (2007). Measure Theory, Vol. 2. Springer, Berlin.Google Scholar
Canay, I. V., Santos, A. and Shaikh, A. M. (2013). On the testability of identification in some nonparametric models with endogeneity. Econometrica 81, 2535–2559.Google Scholar
Carvalho, S. L. and Condori, A. (2021). Generic properties of invariant measures of full-shift systems over perfect Polish metric spaces. Stoch. Dynam. 21, 2150040.Google Scholar
Choksi, J. and Prasad, V. (2006). Approximation and Baire category theorems in ergodic theory. In Measure Theory and its Applications, eds. J.-M. Belley, J. Dubois and P. Morales (Lect. Notes Math. 1033). Springer, New York, pp. 94–113.Google Scholar
Corradi, V., Swanson, N. R. and White, H. (2000). Testing for stationarity–ergodicity and for comovements between nonlinear discrete time Markov processes. J. Econometrics 96, 39–73.Google Scholar
Cover, T. M. and Thomas, J. A. (2012). Elements of Information Theory. Wiley, Chichester.Google Scholar
Domowitz, I. and El-Gamal, M. A. (1993), A consistent test of stationary-ergodicity. Econometric Theory 9, 589–601.Google Scholar
Domowitz, I. and El-Gamal, M. A. (2001). A consistent nonparametric test of ergodicity for time series with applications. J. Econometrics 102, 365–398.Google Scholar
Eisner, T. and Sereny, A. (2009). Category theorems for stable semigroups. Ergodic Theory Dynam. Syst. 29, 487–494.Google Scholar
Friedman, N. A. (1970). Introduction to Ergodic Theory. Van Nostrand Reinhold, New York.Google Scholar
Gelfert, K. and Kwietniak, D. (2018). On density of ergodic measures and generic points. Ergodic Theory Dynam. Syst. 38, 1745–1767.Google Scholar
Gray, R. M., Neuhoff, D. L. and Shields, P. C. (1975). A generalization of Ornstein’s $\bar d$ distance with applications to information theory. Ann. Prob. 3, 315–328.Google Scholar
Grazzini, J. (2012). Analysis of the emergent properties: Stationarity and ergodicity. J. Artificial Soc. Social Sim. 15, 7.Google Scholar
Grillenberger, C. and Krengel, U. (1976). On marginal distributions and isomorphisms of stationary processes. Math. Z. 149, 131–154.Google Scholar
Guerini, M. and Moneta, A. (2017). A method for agent-based models validation. J. Economic Dynam. Control 82, 125–141.Google Scholar
Halmos, P. R. (1944). In general a measure preserving transformation is mixing. Ann. Math. 45, 786–792.Google Scholar
Halmos, P. R. (2017). Lectures on Ergodic Theory. Dover Publications, Mineola, NY.Google Scholar
Hanneke, S. (2021). Learning whenever learning is possible: Universal learning under general stochastic processes. J. Mach. Learn. Res. 22, 1–116.Google Scholar
Iommi, G., Todd, M. and Velozo, A. (2020). Upper semi-continuity of entropy in non-compact settings. Math. Res. Lett. 27, 1055–1077.Google Scholar
Itô, K. (1984). An Introduction to Probability Theory. Cambridge University Press.10.1017/9781139171809CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability. Springer, New York.Google Scholar
Karlin, S. and Taylor, H. E. (2012). A First Course in Stochastic Processes. Elsevier, Amsterdam.Google Scholar
Kieffer, J. C. (1974). On the approximation of stationary measures by periodic and ergodic measures. Ann. Prob. 2, 530–534.Google Scholar
Kieffer, J. C. (1980). On coding a stationary process to achieve a given marginal distribution. Ann. Prob. 8, 131–141.Google Scholar
Kieffer, J. C. (1983). On obtaining a stationary process isomorphic to a given process with a desired distribution. Monatshefte Math. 96, 183–193.Google Scholar
Loch, H., Janczura, J. and Weron, A. (2016). Ergodicity testing using an analytical formula for a dynamical functional of alpha-stable autoregressive fractionally integrated moving average processes. Phys. Rev. E 93, 043317.Google Scholar
Ornstein, D. S. (1973). An application of ergodicy theory to probability theory. Ann. Prob. 1, 43–58.Google Scholar
Ornstein, D. S. and Weiss, B. (1990). How sampling reveals a process. Ann. Prob. 18, 905–930.Google Scholar
Parthasarathy, K. R. (1961). On the category of ergodic measures. Illinois J. Math. 5, 648–656.Google Scholar
Platt, D. (2020). A comparison of economic agent-based model calibration methods. J. Economic Dynam. Control 113, 103859.Google Scholar
Rokhlin, V. A. (1948). A general measure-preserving transformation is not mixing. Dokl. Akad. Nauk 60, 349–351.Google Scholar
Rüschendorf, L. and Sei, T. (2012). On optimal stationary couplings between stationary processes. Electron. J. Prob. 17, 1–20.Google Scholar
Ryabko, D. (2010). Discrimination between b-processes is impossible. J. Theoret. Prob. 23, 565–575.Google Scholar
Shields, P. C. (1996). The Ergodic Theory of Discrete Sample Paths. American Mathematical Society, Providence, RI.Google Scholar
Silva, C. E. (2007). Invitation to Ergodic Theory (Student Math. Library 42). American Mathematical Society, Providence, RI.Google Scholar
Srivastava, S. M. (1998). A Course on Borel Sets. Springer, New York.Google Scholar
Tao, T. (2012). Topics in Random Matrix Theory. American Mathematical Society, Providence, RI.Google Scholar
Walters, P. (2000). An Introduction to Ergodic Theory. Springer, New York.Google Scholar
Wang, H., Wang, C., Zhao, Y. and Lin, X. Toward practical approaches toward ergodicity analysis. Theoret. Applied Climatology 138, 1435–1444.Google Scholar