Abstract
We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.
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References
Ash, R.: Topics in Stochastic Processes. Academic Press, New York (1975)
Bayart, F., Konyagin, S.V., Queffélec, H.: Real Anal. Exch. 33, 1–31 (2004)
Bourgain, J., Kahane, J.P.: Ann. Inst. Fourier 60(4), 1201–1214 (2010)
Dedecker, J., Merlevède, F.: Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30, 1044–1081 (2002)
Duren, P.L.: Theory of H p-Spaces. Dover, New York (2000)
Gordin, M.I.: The central limit theorem for stationary processes. Sov. Math. Dokl. 10, 1174–1176 (1969)
Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. Academic Press, New York (1980)
Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs (1962)
Korevaar, J.: Tauberian Theory, a Century of Developments. Springer, Berlin (2004)
Maxwell, M., Woodroofe, M.: Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28, 713–724 (2000)
Newman, D., Shapiro, H.: The Taylor coefficients of inner functions. Mich. Math. J. 2, 249–255 (1962)
Peligrad, M., Utev, S.: A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33, 798–815 (2005)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
Volný, D.: Determinism and martingale decomposition of strictly stationary sequences. In: Proceedings of the 13th Winter School on Abstract Analysis. Suppl. si Rend. di Circ. Mat. di Palermo, Serie II, Numero 10, pp. 185–192 (1985)
Volný, D.: The central limit problem for composed arrays of martingale differences. Stat. Decis. 10, 281–290 (1992)
Volný, D.: Approximating martingales and the CLT for strictly stationary processes. Stoch. Process. Appl. 44, 41–74 (1993)
Volný, D.: Martingale approximations of stochastic processes depending on the choice of the filtration. An unpublished result (2009)
Wu, W.B.: Nonlinear system theory: another look at dependence. Proc. Natl. Acad. Sci. USA 102, 14150–14154 (2005)
Wu, W.B.: Strong invariance principles for dependent random variables. Ann. Probab. 35, 2294–2320 (2007)
Wu, W.B., Woodroofe, M.: Martingale approximation for sums of stationary processes. Ann. Probab. 32, 1674–1690 (2004)
Zhao, U., Woodroofe, M.: On martingale approximations. Ann. Appl. Probab. 18, 1831–1847 (2008)
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Queffélec, H., Volný, D. On Martingale Approximation of Adapted Processes. J Theor Probab 25, 438–449 (2012). https://doi.org/10.1007/s10959-011-0386-z
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DOI: https://doi.org/10.1007/s10959-011-0386-z