ABSTRACT

Note that the definitions of asymptotic stationarity above only involve the values of Xt for t 0 when G = and for t in the positive cone when G = n. Thus even when n = 1, the behavior of Xt for t < 0 is not taken into account (unless X is stationary), and as n less and less of the nature of X plays a role in the definition, so the function R( ) carries less and less information about the process. Moreover, when G n or n, there is no notion of positivity. (The exception here is the collection of ordered groups considered by Helson and Lowdenslager [26], [27]. For an application of this setting to prediction theory, see the work of Mandrekar and Nadkarni [43].) So we shall adopt the more symmetric notion of an asymptotically stationary process below.