Optimising Prediction Error among Completely Monotone Covariance Sequences

In [9] the aggregation of simple dynamic equations, in particular AR(1) models, was introduced as a means for generating processes displaying long memory. Since this paper, the idea of aggregating has become a popular approach for constructing time series with a flexible covariance structure (see, for instance [1, 12, 2, 19, 15, 17]). Aggregation has also become a useful way to represent certain time series and provides a method for simulation. Examples of time series which can be represented in this way include the fractional ARIMA(0, d, 0) model proposed by Granger and Joyeaux in [10] and Lamperti’s transformation of fractional Brownian motion with H < 1/2 [5]. Other examples are given in [14]. The problem of representing long memory processes in terms of an aggregation of short memory processes is studied in [7]. Consider the aggregation of independent Gaussian AR(1) time series with positive correlation parameters. The covariance sequence of this time series is represented as