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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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