Limiting distributions of time averages for processes with a semi-Markov interference (Q918563)

Let \((x_ t)\) be a stochastic process with an arbitrary state space (X,\({\mathcal B})\) and with semi-Markov interference \(\tau\) [see \textit{V. M. Shurenkov}, Math. USSR, Sb. 54, 161-183 (1986); translation from Mat. Sb., Nov. Ser. 126(168), No.2, 172-193 (1985; Zbl 0595.60029)]. If \(f:(X,{\mathcal B})\to (R_+,{\mathcal B}_+)\) (where \(R_+=[0,\infty)\) and \({\mathcal B}_+\) is the Borel \(\sigma\)-algebra on \(R_+)\) is a Borel measurable function, let be \(\zeta_ t=\int^{t}_{0}f(x_ u)du.\) Conditions for the existence of the limit distribution \(\zeta_ t/t\) are given if the average times \(\tau\) are not finite.