On the distribution function of the complexity of finite sequences

Investigations of complexity of sequences lead to important applications such as effective data compression, testing of randomness, discriminating between information sources and many others. In this paper we establish formulae describing the distribution functions of random variables representing the complexity of finite sequences introduced by Lempel and Ziv in 1976. It is known that this quantity can be used as an estimator of entropy. We show that the distribution functions depend affinely on the probabilities of the so-called “exact” sequences.