An individual ergodic theorem for fuzzy random variables (Q801383)

In probability theory, the strong law of large numbers is just a special case of ergodic theorem. The present authors have proved a strong law of large numbers for Kwakernaak's fuzzy random variables (see the foregoing review, Zbl 0551.60035). This work generalizes it to an individual ergodic theorem. If T is a measure-preserving transformation on a probability space (\(\Omega\),\({\mathcal A},P)\), then, (1) for any frv \(\xi\) on (\(\Omega\),\({\mathcal A},P)\), \(n^{-1}\sum^{n- 1}_{k=0}\xi (T^ k\omega)\) converges a.s., (2) the limit, denoted by \(\xi^*\), is a frv and invariant, i.e., \(\xi^*(T^ n\omega)=\xi^*(\omega)\) a.s. \((n=0,\pm 1,\pm 2,...),\) (3) \(E\xi^*=E\xi,\) (4) if T is an ergodic measure-preserving transformation, \(\xi^*(\omega)=E\xi\) a.s. This is a generalization of Birkhoff's individual ergodic theorem to the fuzzy situation.