By Jacob Wolfowitz
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Additional info for Coding Theorems of Information Theory: Reihe: Wahrscheinlichkeitstheorie und Mathematische Statistik
A}, where q4 depends on n as a parameter, as follows: q4(k / n) a' = l: niq3(k / i). 19) i =1 Obviously the distribution q4 bears the same relation to q3 as n' does to n in Chapter 3. We see that C4= m;x [H(q4(' / n)) -i~ ni H (q3('/ i))]. c. 3. , 0 < J. ;;;; 1, be arbitrary. ) for channel IV. 3 holds (see the remarks at the end of this chapter). 4. For any n a code (n, N, A) for channel IV satisfies logN < + nC4 1 I _ A . 21) The proof, to be given in the next section, will be intelligible only to a reader familiar with the contents of Chapter 7.
6 below. 1) below. 6 as follows: Let So be an arbitrary but fixed state. c. f. is w*. 3). 32 4. 6. In the theory of Markov chains there is a simple standard device for reducing chains of order m > 1 to simple chains (of order one); one uses new states which are m-vectors in the old states. The same device can also be used here. 10) In Chapters 2 and 3 we have assumed that both input and output alphabets have the same number of elements. The case where the input alphabet has more elements than the output alphabet is subsumed under the case of equal numbers.
13) We wish to bound Ie loge - b logbl. Obviously there is difficulty only for small c. Take e ~ I e' Then c loge < b 10gb. 13), b >e- 4a2(1 +6)Vc =q Vn Suppose first that q ~ (say). 14) 0. 15) -~). 16) 4. Compound Channels 38 which is less than a positive constant times a 2 (I + 15) n -to Now suppose that q < O. 12). 18) where K7 is a positive constant. 18) is satisfied by all s e 5* associated with soo. 19) where the maximum is taken over all indices s associated with Soo. 6 we conclude that Ai, i = I, ...
Coding Theorems of Information Theory: Reihe: Wahrscheinlichkeitstheorie und Mathematische Statistik by Jacob Wolfowitz