THE TRANSITION AND AUTOCORRELATION STRUCTURE OF TES PROCESSES Part I: General Theory

TES (Transform-Expand-Sample) is a versatile class of stochastic sequences which can capture arbitrary marginals and a wide variety of sample path behavior and autocorrelation functions. In TES, the initial variate is uniform on [0,1) and the next variate is obtained recursively by taking the fractional part (i.e., modulo-1 reduction) of a linear autoregressive scheme. We show how this class gives rise to uniform Markovian sequences in a general and natural way, by observing that marginal uniformity is closed under modulo-1 addition of an independent variate with arbitrary distribution. We derive the transition function of TES sequences and the autocovariance function of transformed TES sequences using Fourier and Laplace Transform methods. The autocovariance formulas are amenable to fast and accurate calculation and provide the theoretical basis for a computer-based methodology of heuristic TES modeling of empirical data. A companion paper contains various examples which show the efficacy of the TES approach by comparing numerical and simulation-based calculations for a variety of TES autocorrelation functions. The results have applications to the modeling of autocorrelated sequences, particularly in a Monte Carlo simulation context.