Empirical spectral and bispectral analyses of periodically nonstationary stochastic processes
Let \(\xi(t)\), \(t \in R\), be a real periodically nonstationary random process with \(E \xi(t)=0\), \(R(t,\tau)=E \xi(t+\tau)\xi(t)\) and spectral density \[ f_ j (\lambda)=(2 \pi T)^{-1}\int^ T_ 0e^{-ij \gamma t}dt \int^ \infty_{-\infty}e^{-i \lambda \tau} R(t,\tau)d \tau,\;\gamma=2 \pi /T,\;j \in \mathbb{Z}. \] (If all finite-dimensional distributions of random process \(\xi(t)\), \(t \in R\), are invariant about shift of \(t\) on quantity \(T>0\) \((T\) is a period of nonstationarity), then the process \(\xi(t)\), \(t\in R\), is called periodically nonstationary stochastic process.) Let be \(h+(Nh)^{-1} \to 0\), \(N \to \infty\), \(h \in \mathbb{N}\), \(W(\cdot)\) be a weight functions \(\omega \in R\), \[ \begin{aligned} f^{(N)}_ k(\omega) &=(2\pi)^{-1}\int^ \infty_{-\infty} e^{-i\omega \tau}R^{(N)}_ k (\tau)W(h \tau)d \tau,\\ R_ k^{(N)}(\tau)&=(NT)^{-1}\int^{NT}_ 0 e^{-ik \gamma t} \xi(t)\xi(t+\tau)dt,\quad k \in \mathbb{Z}.\end{aligned} \] \(f^{(N)}_ k(\omega)\) is an estimation of \(f_ k(\omega)\). Limiting properties of \(f_ k^{(N)}(\omega)\) are investigated. A similar estimation for the bispectral density is also investigated.
periodically nonstationary random process
scientific article; zbMATH DE number 482597