Remarks on a noncanonical representation for a stationary Gaussian process (Q5937985)

Let \(X=(X(t))_{t\in\mathbb R}\) be a stationary Gaussian process with moving average representation \[ X(t) =\int_{-\infty}^t F(t-u) d B(u) ,\quad t\in\mathbb R. \] Let \(c(\lambda)\) denote the inverse Fourier transform of \(F\), then the Karhunen representation asserts \[ c(\lambda) = C\cdot c_0(\lambda) c_1(\lambda) c_2(\lambda) c_3(\lambda) \] for certain well-described functions \(c_0,c_1,c_2\) and \(c_3\). Here \(c_0\) denotes the outer function of \(c\), while the inner functions \(c_1, c_2\) and \(c_3\) are called the Blaschke part, the delay part and the singular part, respectively. The aim of the present paper is to relate properties of \(c_1\) with those of the linear span \[ H_t(X):= \text{ span}\{ X(s) : s\leq t\} . \] Recall that \(c(\lambda)=C\cdot c_0(\lambda)\), i.e. \(c_1, c_2, c_3\) are trivial, iff there is a Brownian motion \(B\) on \(\mathbb R\) with \(H_t(X)=H_t(B)\), \(t\in\mathbb R\).