Sufficient conditions of standardness for filtrations of stationary processes taking values in a finite space

• Autores: Gaël Ceillier
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 40, Nº. 5, 2012, págs. 1980-2007
• Idioma: inglés
• DOI: 10.1214/11-AOP666
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• Resumen

  • Let X be a stationary process with finite state-space A. Bressaud et al. [Ann. Probab. 34 (2006) 1589–1600] recently provided a sufficient condition for the natural filtration of X to be standard when A has size 2. Their condition involves the conditional laws p(⋅|x) of X0 conditionally on the whole past (Xk)k≤−1=x and controls the strength of the influence of the “old” past of the process on its present X0. It involves the maximal gaps between p(⋅|x) and p(⋅|y) for infinite sequences x and y which coincide on their n last terms. In this paper, we first show that a slightly stronger result holds for any finite state-space. Then, we provide sufficient conditions for standardness based on average gaps instead of maximal gaps.