INCESSANT TRANSMISSION 9/10 



independent, would make possible. If the sequence is of finite 

 length, e.g. five spins of a coin, as in the previous paragraph, the 

 constraint can be identified and treated exactly as in S.7/1 1 . When, 

 however, it is indefinitely long, as is often the case with sequences 

 (whose termination is often arbitrary and irrelevant) we must use 

 some other method, without, however, changing what is essential. 



What the method is can be found by considering how an infinitely 

 long vector can be specified. Clearly such a vector cannot be 

 wholly arbitrary, in components and values, as was the vector in 

 S.3/5, for an infinity of time and paper would be necessary for its 

 writing down. Usually such indefinitely long vectors are specified 

 by some process. First the value of the initial component is given, 

 and then a specified process (a transformation) is applied to generate 

 the further components in succession (like the "integration" of S. 3/9). 



We can now deduce what is necessary if a set of such vectors is 

 to show no constraint. Suppose we build up the set of "no con- 

 straint", and proceed component by component. By S.7/1 2, the 

 first component must take its full range of values; then each of these 

 values must be combined with each of the second component's 

 possible values; and each of these pairs must be combined with each 

 of the third component's possible values ; and so on. The rule is that 

 as each new component is added, all its possible values must occur. 



It will now be seen that the set of vectors with no constraint corres- 

 ponds to the Markov chain that, at each transition, has all the transitions 

 equally probable. (When the probability becomes an actual frequency, 

 lots of chains will occur, thus providing the set of sequences.) Thus, 

 if there are three states possible to each component, the sequences 

 of no constraint will be the set generated by the matrix 



Ex. 1 : The exponential series defines an infinitely long vector with components : 



.y2 X^ a'* 



^ ' ^' T' 2T3' 2.3.4' • ■ '^ 



What transformation generates the series by obtaining each component 

 from that on its left? (Hint: Call the components t\, t2, . . ., etc.; // is 

 the same as ta i.) 



Ex. 2 : Does the series produced by a true die show constraint ? 



Ex. 3: (Continued.) Does the series of Ex. 9/4/3 ? 



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