Abstract
This paper deals with two typical random number generation problems in information theory. One is the source resolvability problem (resolvability problem for short) and the other is the intrinsic randomness problem. In the literature, optimum achievable rates in these two problems with respect to the variational distance as well as the Kullback-Leibler (KL) divergence have already been analyzed. On the other hand, in this study we consider these two problems with respect to f-divergences. The f-divergence is a general non-negative measure between two probabilistic distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. In this paper, we impose some conditions on the function f so as to simplify the analysis, that is, we consider a subclass of f-divergences. Then, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences. Next, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. The second-order optimum achievable rates and optimistic optimum achievable rates have also been investigated.
Highlights
I N THIS paper, we consider two typical fixed-length random number generation problems: the source resolvability problem and the intrinsic randomness problem
One of main contributions of the present paper is to provide the unified viewpoint to the analysis in the resolvability problem with respect to the subclass of f -divergences
In order to show the general formula of the first-order optimum resolvability rate, we introduce the information quantity on the basis of the function f given 0 ≤ ε < f (0): Kf (ε|X)
Summary
I N THIS paper, we consider two typical fixed-length random number generation problems: the source resolvability problem (the resolvability problem for short) and the intrinsic randomness problem. Vembu and Verdú [7] have considered the intrinsic randomness problem with respect to the variational distance as well as the normalized KL divergence and derived general formulas of the first-order optimum achievable rates (cf Han [8]). The analysis of the optimum achievable rate in the resolvability problem (or the intrinsic randomness problem) has been relied on the specified approximation measure. We establish the general formulas of the first- and second-order optimum achievable rates for several important measures that have not been considered yet This is one of significant contributions of this paper.
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