In , Csisza/spl acute/r established the concept of forward /spl beta/-cutoff rate for the error exponent hypothesis testing problem based on independent and identically distributed (i.i.d.) observations. Given /spl beta/0, /spl alpha//spl ne/1. Similarly, for 0</spl beta/<1, Csisza/spl acute/r also established the concept of reverse /spl beta/-cutoff rate for the correct exponent hypothesis testing problem. In this work, we extend Csisza/spl acute/r's results by investigating the forward and reverse /spl beta/-cutoff rates for the hypothesis testing between two arbitrary sources with memory. We demonstrate that the lim inf Re/spl acute/nyi /spl alpha/-divergence rate provides the expression for the forward /spl beta/-cutoff rate. We also show that if the log-likelihood large deviation spectrum admits a limit, then the reverse /spl beta/-cutoff rate equals the liminf /spl alpha/-divergence rate, where /spl alpha/=(1/1-/spl beta/) and 0</spl beta/</spl beta//sub max/, where /spl beta//sub max/ is the largest /spl beta/<1 for which the lim inf (1/1-/spl beta/)-divergence rate is finite. For /spl beta//sub max//spl les//spl beta/<1, we show that the reverse cutoff rate is in general only upper-bounded by the lim inf Re/spl acute/nyi divergence rate. Unlike in , where the alphabet for the source coding cutoff rate problem was assumed to be finite, we assume arbitrary (countable or continuous) source alphabet. We also provide several examples to illustrate our forward and reverse /spl beta/-cutoff rates results and the techniques employed to establish them.

In this work, we provide a computable expression for the Kullback-Leibler divergence rate lim/sub n/spl rarr//spl infin//1/nD(p/sup (n)//spl par/q/sup (n)/) between two time-invariant finite-alphabet Markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p/sup (n)/ and q/sup (n)/, respectively. We illustrate it numerically and examine its rate of convergence. The main tools used to obtain the Kullback-Leibler divergence rate and its rate of convergence are the theory of nonnegative matrices and Perron-Frobenius theory. Similarly, we provide a formula for the Shannon entropy rate lim/sub n/spl rarr//spl infin//1/nH(p/sup (n)/) of Markov sources and examine its rate of convergence.

In an earlier work, Poor and Verdu (1995) established an upper bound for the reliability function of arbitrary single-user discrete-time channels with memory. They also conjectured that their bound is tight for all coding rates. We demonstrate via a counterexample involving memoryless binary erasure channels (BECs) that the Poor-Verdu upper bound is not tight at low rates. We conclude by examining possible improvements to this bound.

In this work, we examine the existence and the computation of the Renyi divergence rate, lim/sub n/spl rarr//spl infin// 1/n D/sub /spl alpha//(p/sup (n)//spl par/q/sup (n)/), between two time-invariant finite-alphabet Markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p/sup (n)/ and q/sup (n)/, respectively. This yields a generalization of a result of Nemetz (1974) where he assumed that the initial probabilities under p/sup (n)/ and q/sup (n)/ are strictly positive. The main tools used to obtain the Renyi divergence rate are the theory of nonnegative matrices and Perron-Frobenius theory. We also provide numerical examples and investigate the limits of the Renyi divergence rate as /spl alpha//spl rarr/1 and as /spl alpha//spl darr/0. Similarly, we provide a formula for the Renyi entropy rate lim/sub n/spl rarr//spl infin// 1/n H/sub /spl alpha//(p/sup (n)/) of Markov sources and examine its limits as /spl alpha//spl rarr/1 and as /spl alpha//spl darr/0. Finally, we briefly provide an application to source coding.