Given a finite alphabet, there is an inductive method for constructing a stationary measure on doubly infinite words from this alphabet. This construction can be randomized; the main focus here is on a particular uniform randomization which intuitively corresponds to the idea of choosing a generic stationary process. It is shown that with probability 1, the random stationary process has zero entropy and gives positive probability to every periodic infinite word.
Kenneth S. Alexander. Steven A. Kalikow. "Random Stationary Processes." Ann. Probab. 20 (3) 1174 - 1198, July, 1992. https://doi.org/10.1214/aop/1176989685