Realization theory for two-time-scale distributions

In this paper we investigate the problem of approximating the Markov parameters of a two-time-scale distribution. It is shown that the Markov parameters of a two-time-scale distribution can be approximated in terms of the Markov parameters of its fast distribution only. This is an approximation, which deteriorates by a factor of 1/ε for every higher order Markov parameter. In order to use the Markov parameters of the slow distribution in the approximation scheme, an inversion map is introduced by which a two-time-scale distribution and its slow distribution are mapped into new distributions. It is then shown that every Markov parameter of the inverted distribution to within an O(ε) quantity. An approximate expression for the Hankel matrix of the Markov parameters of a two-time-scale distribution is also obtained. This expression is in form of a telescopic series and involves the Markov parameters of the fast distribution only.