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The continued fraction as an information source (Corresp.)

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Nearly every digit (positive-integer partial quotient) of the simple continued fraction representing a random number carries 3.4325 bits of information about the number. This is reduced by 0.0081 bit if the preceding digit is already known and by a further 0.0007 bit if the second preceding digit is also known, Any finite sequence of digits is equally likely to occur forward and backward. Hence, the digit (source output symbol) following any digit likewise gives 0.0081 bit of information about the latter, and all succeeding (or preceding) digits together give 0.0088 bit despite the unbounded range of the statistical dependence among digits.

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Information Theory, IEEE Transactions on  (Volume:30 ,  Issue: 4 )