This article is cited in 3 papers

Short Communications

On the Stability of Solutions to Linear Problems for Stationary Processes

Yu. A. Rozanov

Moscow

Abstract: 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.

Received: 16.12.1963

 English version: Theory of Probability and its Applications, 1964, 9:3, 477–479

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