This paper is devoted to study of characteristics of logarithmically asymptotically optimal (LAO) hypotheses testing and identification for a model consisting of two related objects. In considered general case of dependence it is supposed that L1possible probability distributions of states constitute the family of possible hypotheses for the first object and the second object is distributed according to one of L1× L2given conditional distributions depending on the distribution index and the current observed state of the first object. By one approach the matrix of interdependencies of all possible pairs of the error probability exponents (reliabilities) of asymptotically optimal tests of distributions of both objects is studied. The identification of the distributions of two objects gives an answer to the question whether r1-th and r2-th distributions occurred, or not on the first and the second objects, correspondingly. Interdependence of reliabilities for the LAO identification is determined for each pair of double hypotheses. By the second approach the optimal interdependencies of lower estimates of all possible pairs of corresponding reliabilities are found and lower estimates of reliabilities for the LAO identification are studied for all pairs of hypotheses.

Two problems concerning arbitrarily varying stationary Markov source (AVMS), namely, the binary hypothesis testing and the source coding problems are solved. First, we consider a logarithmically asymptotically optimal (LAO) hypothesis testing (HT) for distributions of AVMS. The asymptotic behavior of the second type error probability exponent is investigated in function of the first type error probability exponent, as the number of observations tends to infinity. In the problem of AVMS coding, the E-optimal rate function R(E) (the minimum rate R of the source sequences compression when the decoding error probability is less than exp{-NE}, (E > 0) and its inverse reliability function E(R) are obtained from the corresponding HT result.

In this paper, we survey the concept of the$E$-capacity as a generalization to the Shannon's capacity of a channel and as an counterpart to the Shannon's reliability function. Random coding, expurgated and sphere packing bounds are derived by the method of types and by the method of graph decomposition for$E$-capacity of discrete memoryless channel (DMC). Three decoding rules are considered. This paper summarizes and revises the results of previous hardly accessible publications of the author.

Some important properties of the rate-reliability-distortion function of discrete memoryless source (DMS) are established. For the binary source and Hamming distortion measure this function is derived and analyzed. Even that elementary case suffices to show the nonconvexity of the rate-reliability-distortion function in the reliability (error exponent) argument.

A source coding problem is considered for a robust description system with one encoder and many decoders. The rate-reliability-distortion function is specified. An example is given showing the distinction in the calculation of the rate-reliability-distortion function as compared with the rate-distortion function.

The results of the article "Multiterminal source coding achievable rates and reliability" (IEEE Trans. Inform. Theory, vol. 42, pp. 2094-2101, Nov. 1996) proved to be correct only in very particular cases. The author is grateful to R. Ahlswede and N. Cai for a counterexample, to I. Csiszar as well as T. Berger for letters to the author with remarks on the subject. My persistent and prolonged attempts to find the correct solutions are not successful. The problem remains open.

Some open problems concerning classical multiterminal configuration in which two correlated sources {X} and {Y} are encoded separately and decoded by common decoder with respect to a fidelity criterion for messages of both sources are resolved. The results of Berger and Yeung (1989) are expanded in two directions. First, we determine the "rate-distortion" region R(/spl Delta/) of achievable rates R/sub x/, R/sub y/ for a given /spl Delta/=(/spl Delta//sub x/,/spl Delta//sub y/) with /spl Delta//sub x/ the permissible distortion level for reproduction of X and /spl Delta//sub y/- for Y. Thus we give a full solution of the problem noted by Kaspi and Berger (1982) (they constructed the inner bound for R(/spl Delta/)), and solved by Berger and Yeung for the case of "one distortion". Second, we introduce the notion of "rate-reliability, distortion" region for the multiterminal source encoding problem; more precisely, the region of admissible rates of codes ensuring exponential decrease with a given exponent E>0 of the probability of exceeding the distortion levels /spl Delta//sub x/ or /spl Delta//sub y/. The inner and outer bounds for this region R(E,/spl Delta/) are constructed. The analytical representation obtained for R(/spl Delta/) proved to be somewhat simpler than that considered by Kaspi and Berger and by Berger and Yeung. Ours uses only variables pertaining to the problem statement with no auxiliary ones.