Weak variable-length source coding

Given a general source X={X/sup n/}/sub n=1//sup /spl infin//, source coding is characterized by a pair (/spl phi//sub n/, /spl psi//sub n/) of encoder /spl phi//sub n/, and decoder /spl psi//sub n/, together with the probability of error /spl epsi//sub n//spl equiv/Pr{/spl psi//sub n/(/spl phi//sub n/(X/sup n/))/spl ne/X/sup n/}. If the length of the encoder output /spl phi//sub n/(X/sup n/) is fixed, then it is called fixed-length source coding, while if the length of the encoder output /spl phi//sub n/(X/sup n/) is variable, then it is called variable-length source coding. Usually, in the context of fixed-length source coding the probability of error /spl epsi//sub n/ is required to asymptotically vanish (i.e., lim/sub n/spl rarr//spl infin///spl epsi//sub n/=0), whereas in the context of variable-length source coding the probability of error /spl epsi//sub n/ is required to be exactly zero (i.e., /spl epsi//sub n/=0/spl forall/n=1, 2, ...). In contrast to these, we consider the problem of variable-length source coding with asymptotically vanishing probability of error (i.e., lim/sub n/spl rarr//spl infin///spl epsi//sub n/=0), and establish several fundamental theorems on this new subject.