Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $(X_t)_{t\in\mathbb{Z}}$ be a real-valued second-order process with $X_t\in L^2(\Omega,\mathcal{F},\mathbb{P})$ for every $t$. For each $t\in\mathbb{Z}$, let $\mathcal{R}_t$ be a closed linear subspace of $L^2(\Omega,\mathcal{F},\mathbb{P})$, interpreted as the remote linear past at time $t$. Suppose that a Wold-type decomposition is defined by \begin{align*} D_t=P_{\mathcal{R}_t}X_t,\qquad Y_t=X_t-D_t, \end{align*} where $P_{\mathcal{R}_t}$ is the [orthogonal projection](/theorems/437) onto $\mathcal{R}_t$. If another decomposition \begin{align*} X_t=D'_t+Y'_t \end{align*} satisfies $D'_t\in\mathcal{R}_t$ and $Y'_t\perp\mathcal{R}_t$ in $L^2(\Omega,\mathcal{F},\mathbb{P})$ for every $t$, then \begin{align*} D'_t=D_t,\qquad Y'_t=Y_t \end{align*} in $L^2(\Omega,\mathcal{F},\mathbb{P})$ for every $t$. Hence the deterministic component and the purely nondeterministic component are uniquely determined in $L^2$ by $(X_t)$ and the remote-past subspaces $(\mathcal{R}_t)$.