An ergodic-type theorem for generalized Ornstein-Uhlenbeck processes (Q2854184)

The work is concerned with \(\mathbb R^d\)-valued càdlàg radom process \(\xi\) given on a common probability space \((\Omega, \mathcal F, P)\). Let \(\mathcal F^0\) be a sub-\(\sigma\)-algebra of \(\mathcal F\), and \(\nu_0^+\) be the class of all increasing from zero numeral random processes whose values at all times are \(\mathcal F^0\)-measurable random variables. The goal of the article is to find out what assumptions about \(F\in\nu^+_0\) and \(\mathbb R^d\)-valued measurable random process \(\xi\) guarantee that the relation NEWLINE\[NEWLINE \frac1t\int^t_0h(\xi(s-))\,dF(s)-\frac1t\int^t_0\operatorname E(h(\xi(s-))|\mathcal F^0)\,dF(s)\overset {P}\longrightarrow0\text{ as }{t}\infty NEWLINE\]NEWLINE holds for every bounded continuous function \(h : \mathbb R^d\longrightarrow\mathbb C\).