On weak asymptotic isomorphy of memoryless correlated sources (Q1097237)

Let \(\{(X_ i,Z_ i)\}\) be an i.i.d. sequence of random pairs in a finite set \({\mathcal X}\times {\mathcal Z}\); we will call it a discrete memoryless stationary correlated (DMSC) source with generic distribution \(dist(X_ 1,Z_ 1)\). Two DMSC sources \(\{(X_ i,Z_ i)\}\) and \(\{(X_ i',Z_ i')\}\) are called asymptotically isomorphic in the weak sense if for every \(\epsilon >0\) and sufficiently large n, there exists a joint distribution \(dist(X^ n,\quad Z^ n,\quad X^{'n},\quad Z^{'n})\) of n-length blocks of the two sources such that \[ \frac{1}{n}H(X^ n| X^{'n})<\epsilon,\quad \frac{1}{n}H(Z^ n| Z^{'n})<\epsilon,\quad \frac{1}{n}H(X^{'n}| X^ n)<\epsilon,\quad \frac{1}{n}H(Z^{'n}| Z^ n)<\epsilon. \] For single sources of equal entropy, McMillan's theorem implies asymptotic isomorphy in the sense suggested by this definition. For correlated sources, however, no nontrivial cases of weak asymptotic isomorphy are known. We show that some spectral properties of the generic distributions are invariant for weak asymptotic isomorphy, as. The method is easily implementable, and yields normal fuzzy sets, without widening of the resulting function value set.