Šujan, Š. Sinai’s theorem and entropy compression. (English) Zbl 0559.94007 Probl. Control Inf. Theory 12, 419-428 (1983). The author proves a strengthening of Sinai’s theorem to the effect that if a stationary ergodic process \(X=\{X_ i\}\) is \(\bar d-\)close to an IID process of no greater entropy rate, then there is a stationary coding \(Y=\{Y_ i\}\) of X whose distribution is that of the IID process and for which \(\Pr ob[X_ 0\neq Y_ 0]\) is small. The proof uses the Ornstein technique involving ”gadgets”. Alternatively the result may be deduced from a result of the reviewer [Ann. Probab. 12, 204-211 (1984; Zbl 0551.28023), Theorem 1]. Using his result, the author deduces that the following two distortion-rate functions for the process X are identical (where ”h” denotes entropy rate): \[ D_ X^{(1)}(R)=\inf \{\bar d(X,Y):\quad Y\quad IID,\quad h(Y)\leq R\}, \]\[ D_ X^{(2)}(R)=\inf \{\Pr ob[X_ 0\neq Y_ 0]:\quad Y\quad a\quad stationary\quad coding\quad of\quad X,\quad Y\quad IID,\quad h(Y)\leq R\}. \] Reviewer: J.C.Kieffer Cited in 1 Document MSC: 94A29 Source coding 94A34 Rate-distortion theory in information and communication theory 60G10 Stationary stochastic processes Keywords:stationary ergodic process; IID process; stationary coding; distortion- rate functions Citations:Zbl 0551.28023 PDFBibTeX XMLCite \textit{Š. Šujan}, Probl. Control Inf. Theory 12, 419--428 (1983; Zbl 0559.94007)