On subordinated processes

T. N. Siraya

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 131-136

Résumé

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.