On the Stability of Solutions to Linear Problems for Stationary Processes

Yu. A. Rozanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 3, pp. 528-530

Résumé

Let $\xi(t)$ be a stationary process with spectral function $F(\lambda)$, prediction error $$ \sigma^2=\inf\int\left|e^{i\lambda\tau}-\sum_{t\in T}c(t)e^{i\lambda t}\right|^2dF(\lambda) $$ and let $$ \delta(G)^2=\inf\int\left|e^{i\lambda\tau}-\sum_{t\in T}c(t)e^{i\lambda t}\right|^2dF_1(\lambda), $$ where $F_1(\lambda)=F(\lambda)+G(\lambda)$, $dG(\lambda)\geqq 0$ and $\int{dG(\lambda)\leqq h^2}$. Then $\lim\limits_{h\to 0}\sup\limits_G\delta(G)=\sigma$. Other linear problems similar to the prediction one have solutions with the same properties.