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Short Communications
On subordinated processes
T. N. Siraya Leningrad
Abstract:
Second order processes
$x(t)$,
$y(t)$ (
$t\in T\subset R^1$) are considered as curves in the Hilbert space $\mathscr H=\{\xi\colon \mathbf E\xi=0,\mathbf E|\xi|^2<\infty\}$. The process
$y(t)$ is subordinated to
$x(t)$ if
$H(y)\subset H(x)$, where
$H(x)\subset \mathscr H$ is the closed linear span of the random variables
$x(t)$,
$t\in T$.
Theorem 1. {\it Let processes
$x(t)$ and
$y(t)$,
$t\in T$, have correlation functions
$R(s,t)$ and
$B(s,t)$, and
$\Phi(s,t)=\mathbf Ex(t)\overline{y(t)}$ be their cross-correlation function.
The process
$y(t)$ is subordinated to
$x(t)$, if and only if the functions
$F_t=\overline{\Phi(\cdot,t)}$ belong to the Hilbert space
$H(R)$ with the reproducing kernel
$R(s,t)$, and their scalar products in
$H(R)$ are
$\langle F_s,F_t\rangle_R=B(s,t)$.}
An analogous result holds for generalized processes.
Representations of a process as the sum of two orthogonal processes, subordinated to it, are also considered.
Received: 12.01.1976