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Sinai’s theorem and entropy compression. (English) Zbl 0559.94007

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

MSC:

94A29 Source coding
94A34 Rate-distortion theory in information and communication theory
60G10 Stationary stochastic processes

Citations:

Zbl 0551.28023
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