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so that

:

and so

:

and

:

and so by Kraft's inequality there exists a prefix-free code having those word lengths. Thus the minimal satisfies

:

Extension to non-stationary independent sources

Fixed rate lossless source coding for discrete time non-stationary independent sources

Define typical set as:

:

Then, for given , for large enough, . Now we just encode the sequences in the typical set, and usual methods in source coding show that the

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