By Raymond W. Yeung (auth.)

A First path in info concept is an up to date advent to info concept. as well as the classical themes mentioned, it offers the 1st entire therapy of the speculation of I-Measure, community coding idea, Shannon and non-Shannon kind details inequalities, and a relation among entropy and staff concept. ITIP, a software program package deal for proving details inequalities, can also be incorporated. With loads of examples, illustrations, and unique difficulties, this ebook is great as a textbook or reference publication for a senior or graduate point direction at the topic, in addition to a reference for researchers in comparable fields.

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**Example text**

Moreover, D{ ·II·) does not satisfy the triangular inequality (see Problem 10). In the rest of the book, informational divergence will be referred to as divergence for brevity. Before we prove that divergence is always nonnegative, we first establish the following simple but important inequality called the fun damental inequality in information theory. 88) if and only if a = 1. Proof Let f{a) = Ina - a + 1. Then f'{a) = Y]« - 1 and f"{a) = -1/a 2 . Since f(l) = 0,1'(1) = 0, and 1"(1) = -1 < 0, we see that f(a) attains its maximum value 0 when a = 1.

46 whose entropy is infinity. Let P; = Pr{X =I X} = Pr{Z = I}. 177) (1 - Pe) · 0 + P, . 179) > O. Therefore, H(X JX) does not tend to 0 as P, ~ O. ENTROPY RATE OF STATIONARY SOURCE In the previous sections, we have discussed various properties of the entropy of a finite collection of random variables. In this section. we discuss the entropy rate entropy rate of a discrete-time information source . A discrete-time information source {Xk» k ~ I} is an infinite collection of random variables indexed by the set of positive integers.

Then bn -t a as n -t 00. 00 Proof The idea of the lemma is the following . If an -t a as n -t 00, then the average of the first n terms in {ak}, namely bn, also tends to a as n -t 00. The lemma is formally proved as follows . Since an -t a as n -t 00, for every e > 0, there exists N(f) such that Ian - c] < f for all n > N(f) . L lai - al + E. (2. I98) The first term tends to 0 as n --+ 00 . Therefore, for any E > 0, by taking n to be sufficiently large, we can make Ibn - al < 2E. Hence bn --+ a as n --+ 00 , proving the lemma .