On a class of asymptotically stationary harmonizable processes
Asymptotic stationarity of nonstationary second order random processes X(t), \(t\in {\mathbb{R}}\), is considered, that is, conditions for the existence of the limits \[ r(h)=\lim_{t\to \infty}t^{- 1}\int^{t}_{0}E(X(s+h)\overline{X(s)})ds,\quad h\in {\mathbb{R}}, \] are discussed. It is shown that any weakly harmonizable second order random process X(t), \(t\in {\mathbb{R}}\), whose covariance kernel admits a representation of the form \[ EX(s)\overline{X(t)}=\int_{{\mathbb{R}}\times {\mathbb{R}}}e^{i(s\lambda -t\mu)}dB(\lambda,\mu),\quad s,t\in {\mathbb{R}}, \] with a \(\sigma\)-finite spectral bimeasure B, is asymptotically stationary; extending a result by \textit{Yu. A. Rozanov} [Teor. Veroyatn. Primen. 4, 291-310 (1959; Zbl 0089.326)] stating that any strongly harmonizable X(t), \(t\in {\mathbb{R}}\), is asymptotically stationary. It is also shown, by using an example by the reviewer [Ann. Acad. Sci. Fennicae, Ser. A I 591 (1975; Zbl 0307.60034)] that there exist weakly harmonizable random processes with a \(\sigma\)-finite spectral bimeasure, which are not strongly harmonizable.
Asymptotic stationarity of nonstationary second order random processes
scientific article; zbMATH DE number 4005253