Central limit theorem for homogeneous processes with independent increments and semi-Markov commutations
A mixed process with independent increments \((Y(t))_{t\geq 0}\) is considered. The cumulant of the process is a stepped semi-Markov process. The family \((\alpha_{\epsilon})_{\epsilon>0}\) of derived processes is investigated where \(\alpha_{\epsilon}(t)=Y(t/\epsilon)\sqrt{\epsilon}- \Lambda t\sqrt{\epsilon}\), \(\Lambda =m^{- 1}\int_{E}m(x)\lambda(x)\rho(dx)\), \(m=\int_{E}m(x)\rho(dx)\), \(m(x)=\int^{\infty}_{0}tG_ x(dt)\), \(G_ x\) is the distribution function of the first exit time from the state \(x\in E\) of the control semi-Markov process, \(\rho\) is the stationary distribution of the embedded Markov chain and \(\lambda\) (x) is the conditional cumulant of the process Y given the state x of the control semi-Markov process. Some conditions in terms of the control semi-Markov process and conditional cumulants are proved to be sufficient for finite dimensional distributions of the limit process \(\alpha =\lim_{\epsilon \to 0}\alpha_{\epsilon}\) to be normal.
scientific article; zbMATH DE number 3858134