ABSTRACT
We introduce the problem of variable-length (VL) source resolvability, in which a given target probability distribution is approximated by encoding a VL uniform random number, and the asymptotically minimum average length rate of the uniform random number, called the VL resolvability, is investigated. We first analyze the VL resolvability with the variational distance as an approximation measure. Next, we investigate the case under the divergence as an approximation measure. When the asymptotically exact approximation is required, it is shown that the resolvability under two kinds of approximation measures coincides. We then extend the analysis to the case of channel resolvability, where the target distribution is the output distribution via a general channel due to a fixed general source as an input. The obtained characterization of channel resolvability is fully general in the sense that, when the channel is just an identity mapping, it reduces to general formulas for source resolvability. We also analyze the second-order VL resolvability.
ABSTRACT
The first- and second-order optimum achievable exponents in the simple hypothesis testing problem are investigated. The optimum achievable exponent for type II error probability, under the constraint that the type I error probability is allowed asymptotically up to ε , is called the ε -optimum exponent. In this paper, we first give the second-order ε -optimum exponent in the case where the null hypothesis and alternative hypothesis are a mixed memoryless source and a stationary memoryless source, respectively. We next generalize this setting to the case where the alternative hypothesis is also a mixed memoryless source. Secondly, we address the first-order ε -optimum exponent in this setting. In addition, an extension of our results to the more general setting such as hypothesis testing with mixed general source and a relationship with the general compound hypothesis testing problem are also discussed.
