Source codes as random number generators

A random number generator generates fair coin flips by processing deterministically an arbitrary source of nonideal randomness. Pin optimal random number generator generates asymptotically Fair coin flips from a stationary ergodic source at a rate of bits per source symbol equal to the entropy rate of the source, Since optimal noiseless data compression codes produce incompressible outputs, it is natural to investigate their capabilities as optimal random number generators. In this paper we show under general conditions that optimal variable-length source codes asymptotically achieve optimal variable-length random bit generation in a rather strong sense, In particular, we show in a-hat sense the Lempel-Ziv algorithm earn be considered an optimal universal random bit generator front arbitrary stationary ergodic random sources with unknown distributions.