Universal Types and Simulation of Individual Sequences

Abstract

We define the universal type class of an individual sequence x $_{\rm 1}^{n}$, in analogy to the classical notion used in the method of types of information theory. Two sequences of the same length are said to be of the same universal (LZ) type if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978). We show that the empirical probability distributions of any finite order k of two sequences of the same universal type converge, in the variational sense, as the sequence length increases. Consequently, the logarithms of the probabilities assigned by any k-th order probability assignment to two sequences of the same universal type converge, for any k. We estimate the size of a universal type class, and show that its behavior parallels that of the conventional counterpart, with the LZ78 code length playing the role of the empirical entropy. We present efficient procedures for enumerating the sequences in a universal type class, and for drawing a sequence from the class with uniform probability. As an application, we consider the problem of universal simulation of individual sequences. A sequence drawn with uniform probability from the universal type class of x $_{\rm 1}^{n}$ is a good simulation of x $_{\rm 1}^{n}$ in a well defined mathematical sense.