Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformity distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An m^k-valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k=2 being very distinctly non-linear and k=10 being virtually indistinguishable from independence.