Some remarks on stationary possibilistic processes.
Hugo Janssen, Gert de Cooman, Etienne E. Kerre · 1998
We investigate the following extendability problem for systems, for which the available information is given by a monotone set mapping M on the field CT of measurable cylinders of a product ample space (XT, RT): given that M is invariant under a RT − RT- measurable transformation H of XT, i.e. M(H−1 (B)) = M(B) for all B ∈ CT, is it possible to find H-invariant monotone extensions of M to the powerclass of XT? We first show that the outer and inner measures of M always have the desired invariance property. If the system that we are dealing with is possibilistic, a number of sufficient conditions are derived to ensure the H-invariance of the greatest possibilistic extension Π g M of M. Consequently stationary possibilistic processes can be represented by a shift-invariant possibility measure on their basic space. As an illustration for our results, we show that possibilistic Markov processes with stationary transition possibilities and stationary initial possibilities are stationary processes. 1