By William Feller

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36) is determined 35 - by C(~,TI), 9 9 9 ,C(~,Tq). 37) the points of D* as a I < a 2 < ... < aZ(k)_l. Also write %0 = i, aZ(k) = i. 35) may be a different C(~, ~ ) a z < ~ < az+ I. in the interval depends upon in ~ for any ~ , for different Then clearly (az,az+l), because in the interval. 35) Suppose intervals (az,az+l) ; there interval. the different are monotonic However, since right members for any C, and thus are totally ordered. 34). (C) the first bracket on the in it are determined by the of = C(~,C) in terms of the C's of ~ .

30) does not depend on g. (It may, and actually does, depend on the fixed ~. 36) is determined 35 - by C(~,TI), 9 9 9 ,C(~,Tq). 37) the points of D* as a I < a 2 < ... < aZ(k)_l. Also write %0 = i, aZ(k) = i. 35) may be a different C(~, ~ ) a z < ~ < az+ I. in the interval depends upon in ~ for any ~ , for different Then clearly (az,az+l), because in the interval. 35) Suppose intervals (az,az+l) ; there interval. the different are monotonic However, since right members for any C, and thus are totally ordered.

15) a). is therefore unique. 15) be t. 17) be a probability We define t = i - max rain ~ ~i 8(s,i). ), 1 lira ~ log N(n,W) > Csr... m n-~oo Let ~* be the maximizing enough. t-vector. 16). For suitable re(e) > 0 we construct Let c > 0 be small a code (n,N = exP2 [n Cgr.. ,U. Consider the new code , the sender's partial ... )}. as i = 1,. ,N. knowledge - SO - (u 9 i) , Ai ) , i : l , . . , ~ . ~ 9 i (u. t ) , a i ) , 9 9 9 9 9 9 i = N- (~_ + . . + 9 9 " %_l 9 9 ) + 1, . . ,N. ,k the average error of this code is not greater than t i- .

### An Introduction to Probability Theory and its Applications by William Feller

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