On the Isomorphism Problem for Stationary Processes
A. Kh. Zaslavskii · Theory of Probability and Its Applications · 1964
The central problem in ergodic theory is that of isomorphism. In the paper the sufficient condition for isomorphism of the stationary process $\xi = ( \cdots ,\xi _{ - 1} ,\xi _0 ,\xi _1 , \cdots )$, $\xi _n = 0$, $1, \cdots ,l$, with some stationary process $\eta = ( \cdots ,\eta _{ - 1} ,\eta _0 ,\eta _1 , \cdots )$, $\eta _n = \alpha _1 , \cdots ,\alpha _m $, $m \leqq l$, is found. This condition is expressed in terms of a one-dimensional distribution of the process $\xi $. Isomorphism is constructed with the aid of elementary codes \[ (i) = \eta _1^i \eta _2^i \cdots \eta _{\omega _i }^i ,\qquad i = 1, \cdots ,l, \] which code the elementary words \[ (i) = \underbrace {i00 \cdots 0.}_{\omega _i } \] One of the examples considered proves that it is possible to construct a system of elementary codes for any arbitrary l and m. This system possesses some properties which secure unique decoding of the sequence $\eta $.