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13) i 2 i But H(-Jt-oJt): H(l'1t- "t21 , ~ign(vt- vt)) i 2\\ 2 f ~ H('-'t- Vt) + 1 (cf. 2 . 11), cf. 4). Corollary 1. 15) Chapter 3. m l_t H(~..... 16) holds. •• Then provided that "rn v: = [t] and, converges in probability to q which is the same n" as saying that~ converges in probability to 1, we have t1 Vqt1 - Vt2 0. E... 16). 17) t. For uniformly integrable families of random variables, convergence in probability does imply convergence of the mathematical expectation, a fact that enables us to assert Corollary 2.

25), it is in order to ernphasize that the average code length per letter 9 was here assurned to exist in a stronger sense that in section 2. In fact, not only the e:xpecta tion of required to converge to q , but II g (~ 1 ••• ~n)jj divided by n was II g (~" ... ~ 11 ) II itself (in proba- bility). Mathernatica lly, this is a significant difference, but, practically , the two conditions are usually fulfilled sirnultaneously. CONTENTS Chapter 1. The simplest coding problern ................... Page 6 Chapter 2.

20) while one always has (provided tha t the entropy ra tes in question 34 Chapter 3. The principle of conservation of entropy exists) H(X II z) ~ H(YII z'). 21) Proof. 19) follows ~~ = ~i . . 21). If there are at most m sequences ueX with the same g(u) then H(i~)=H(~~,gC~t»=H(fti9Cit~+ + H( g (~ t)) , where the tenn H(~tl g (~ Thus, in view of g(~~) =i;, t)) is upper bounded by Pog m . 20). Lemma 2 is a first (trivial) way of stating the principle of conservation of entropy. A more interesting result is obtained by combining this result with theorem 1.

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