# Ergodic and Prediction Problems

• Murray Rosenblatt
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 184)

## Summary

In this chapter we will be concerned with the asymptotic behavior of Markov processes with stationary transition probability function P(·,·). It is occasionally useful to look at the process only when it falls within a proper measurable subset A of the state space Ω. In the first section, conditions under which a well-defined “process on A” can be defined will be established. Sometimes it is useful to look at such derived processes when trying to establish the existence of subinvariant or invariant measures μ, that is, σ-finite measures for which
$$\int {\mu (dx)P(x,A) \leqslant \mu (A)}$$
or
$$\int {\mu (dx)P(x,A) = \mu (A)}$$
hold for each set A ∈ A, respectively. The primary object of this chapter is the study of the asymptotic behavior of partial sums
$$\sum\limits_{j = 1}^n {{P_j}} ( \cdot , \cdot )$$
(1)
or of
$${P_n}( \cdot , \cdot )$$
(2)
itself as n→∞. Problems dealing with the partial sums (1) are related to ergodic theory while those concerned with the transition probability function (2) itself are more closely related to prediction. In discussions of ergodic and prediction problems, the existence of an invariant (or subinvariant) measure is usually required. Some conditions under which such a measure exists will be derived in section 3 under assumptions of a topological character. However, in discussions of ergodic and prediction problems the existence of such a measure will be taken for granted. In section 2 an ergodic theorem due to Chacon and Ornstein will be given with the classic Birkhoff ergodic theorem for stationary processes as an almost immediate corollary. Some aspects of a least squares version of the (nonlinear) prediction problem for a Markov process are sketched in section 4. It is quite clear that they are directly related to the asymptotic behavior of the transition function or equivalently to the asymp­totic behavior of powers of the induced operator.

Assure fEf2