Nonlinear Representations in Terms of Independent Random Variables

Summary

This chapter is concerned with the question of one-sided nonlinear representations of Markov sequences in terms of independent random variables. Such representations could be phrased in the broader context of stationary (possibly non-Markovian) processes and our investigation in the Markovian context is a question of convenience. One way of motivating such problems is by considering the linear Wold representation of a weakly stationary sequence. The Wold representation is briefly derived in section 1. It arises in the context of the linear prediction problem and in the purely nondeterministic case gives a linear resolution of the process in terms of orthonormal random variables. Section 2 describes how the nonlinear problem is suggested by Wold’s result if linearity is replaced by nonlinearity and orthogonality by independence. Wiener regarded such a nonlinear representation as providing a reencoding of the process. The following section indicates that such a representation can be valid if and only if the process is purely nondeterministic in a nonlinear sense and also poses the question of invertibility of such a representation. The existence of such one-sided nonlinear representations is examined at some length for finite state Markov chains in section 4. Partial results for the more general real-valued Markov process are derived in the last section.