Non-asymptotic and asymptotic analyses on Markov chains in several problems

Abstract

Abstract-In this paper, we derive non-asymptotic achievability and converse bounds on the source coding with side-information and the random number generation with side-information. Our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length. We also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds, which implies that our bounds are asymptotically tight in those regimes. We also show the second order rates of those problems, and derive single letter forms of the variances characterizing the second order rates.

Information geometry approach to parameter estimation in Markov chains

“…Hence, in order to treat the relative entropy D(W V ) and the relative Rényi entropy D 1+s (W V ) in a unified way, we adopt the definition (3.2) for the relative entropy D(W V ) instead of the final term of (5.5). Our definition clarifies the relation between the relative entropy D(W V ) and the relative Rényi entropy D 1+s (W V ), which is helpful when we apply these quantities to simple hypothesis testing [17], random number generation, data compression, and channel coding [18] in Markov chain.…”

“…However, they did not clearly consider the relation with the other conditions in Lemma 4.1. In fact, these equivalence relations are essential for the condition of a generator of an exponential family and also for applications to finite-length evaluations of the tail probability, the error probability in simple hypothesis testing [17], source coding, channel coding, and random number generation [18] in Markov chain. Now, we proceed to the definition of an exponential family for transition matrices.…”

“…We firstly define the potential function φ( θ) from a given transition matrix W and a given generator {g j } Then, we give the parametric transition matrices although their papers [5,2] gave the parametric transition matrices firstly. The potential function φ( θ) for a transition matrix W and a generator {g j } produces several information quantities, which play the central roles when we apply the exponential family for transition matrices to finite-length evaluations of the tail probability and the above applications [17,18] in Markov chain. To characterize these information quantities, we employ an exponential family of transition matrices.…”

“…Also, as discussed in [17], the relative Rényi entropy D 1+s (W V ) plays a central role in simple hypothesis testing as well as the relative entropy D(W V ). Further, these information quantities play an central role in random number generation, data compression, and channel coding [18]. In Section 3, we also give their properties that are useful in the above applications.…”

“…Since the information geometrical structure for probability distributions plays important roles in several topics in information theory as well as statistics, it is better to describe the information geometry of transition matrices so that it can be easily applied to these topics. In fact, the authors applied it to finite-length evaluations of the tail probability, the error probability in simple hypothesis testing, source coding, channel coding, and random number generation in Markov chain as well as the estimation error of parametric family of transition matrices [17,18]. Thus, we revisit the exponential family of transition matrices [2,5] in a manner consistent with the above purpose by using Bregmann divergence [21,20].…”