By J. Wolfowitz (auth.)

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**Example text**

4) instead. Although this is a weaker theorem its proof is of interest because its techniques can be applied to many other channels, and it therefore seems worthwhile to give it here. For a discussion of the difference between a weak and a strong converse see Chapter 7 below. 4. For any n a code (n, N, A) for channel IV satisfies log N nC, + 1 < 1=-T. 21) The proof, to be given in the next section, will be intelligible only to a reader familiar with the contents of Chapter 7. 7 later. 7. 4. c. c.

4. In the next section we prove a strong converse. 9. c. with feedback. 1. 8 satisfies N < 2nc +KY -;;, where K depends on A but not on n. 1) 52 4. Compound Channels We shall first need a preliminary lemma. c. 1. 12). Let Z(k), k = 1, ... , a, be a chance variable with the distribution P{Z(k) = j} = w(j I k), j = 1, ... , a. , instead of writing Z(k) as a subscript to n' we write it as an argument for typographical simplicity). Now R(n) = - i~j~ n;w(j Ii) log ;~ njw(j I i) a a a a ( a ) + 2: 2: n; w (j I i) log w (j I i) .

I . I s) be any c. , not necessarily in 5, and let w (. 1 . f. {V (uol eA} ! 2) 36 4. Compound Channeis where a" ~ 0 as n ~ 00, and a" depends only on band n and not on uo, A, or w (. 1. 1s). 3) n-2 • Then the probability, when any given sequence U o is sent, that any specified sequence V o will be received, is not less than n- 2" under w (,1,1 s), and not less than (n- 2 - a . 2- V;-)" under w (. 1 . 1 s'). We have (n- 2 - a . 2- Yn )" as n~oo. 4). 3) holds. If w Uo 1 i o 1 s) < n- 2 then, for any uo, Ps {N(io,iol uo,v(uo))>l};;;;; !