On the Extrapolation of Generalized Stationary Random Processes
Yu. A. Rozanov · Theory of Probability and Its Applications · 1959
Let D denote the space of all infinitely differentiable functions $\varphi $ that are equal to zero outside an interval; $\xi (\varphi )$ is a random distribution, H is a linear closure of random variables, $\xi (\varphi ) \cdot \varphi \in D$; and $H_S^ - $ is a linear closure of random variables ($\xi (\varphi)$ (where $\varphi \in D$ and $\varphi (t) = 0$ for $t \geqq S$). The random distribution $\xi (\varphi )$ is called singular if $H = \cap _S H_S^ - $ and regular if $ \cap _S = H_S^ - = 0$. The necessary and sufficient conditions for singularity (resp. regularity) of the random distribution $\xi (\varphi )$ are given. The problem of extrapolation is solved for the case where $\xi (\varphi )$ is regular.