Asymptotics for prediction errors of stationary processes with reflection positivity
An \(L_2\) error of prediction for values \(X_T\) of a real, centered, weakly stationary process \(X_s\), \(s\in R\), by observations \(X_s\in[-t,0]\) is considered. The error is defined as \[ V_T(t)=\|P^\bot_{[-t,0]}X_t\|^2-\|P^\bot_{(-\infty,0]}X_t\|^2, \] where \(P^\bot_{[a,b]}X_t\) is an orthogonal complement in \(L_2\) of \(\{X_s,s\in[a,b]\}\) to \(X_T\). It is supposed that the covariance \(R(t)\) of \(X_t\) is regularly varying with negative index: \(R(\lambda t)/R(t)\to\lambda^{-p}\) as \(t\to\infty\) and \(R\) is reflection positive, i.e., \( R(t)=\int_0^\infty e^{-|t|\lambda}\sigma(d\lambda) \) for some finite Borel measure \(\sigma\). Asymptotic formulae for \(V_T(t)\) as \(t\to\infty\) are derived for different \(p\). E.g., if \(0<p<1\) then \[ V_T(t)\simeq (1-p)^2/4t \left( \int_0^T C(s)ds \right)^2, \] where \(C\) is the canonical representation kernel of \(X_t\), i.e., \(R(t)=\int_0^\infty C(t+s)C(s)ds\).
scientific article; zbMATH DE number 1539411