On Isomorphism Problem of Stationary Processes

A. H. Zaslavskiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 318-326

Résumé

The central problem in ergodic theory is that of isomorphism. In the paper the sufficient condition for isomorphism of the stationary process $\xi=(\dots,\xi_{-1},\xi_0,\xi_1,\dots)$, $\xi_n=0$, $1,\dots,l$, with some stationary process $\eta=(\dots,\eta_{-1},\eta_0,\eta_1,\dots)$, $\eta_n=\alpha_1,\dots,\alpha_m$, $m\leqq l$, 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 $$ (i)=\eta_1^i\eta_2^i\cdots\eta_{\omega_i}^i,\qquad i=1,\dots,l, $$ which code the elementary words $$ (i)=\underbrace{i00\dots 0}_{\omega_i}. $$ 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 $\eta$.