Hilbert spaces formed by strongly harmonizable stable processes (Q2726706)

Let \(X= \{X(t), t\in{\mathbf R}\}\) be a strongly harmonizable symmetric \(\alpha\)-stable process which is the Fourier transform of a S\(\alpha X\)S random measure with independent increments \(\Phi\), NEWLINE\[NEWLINEX(t)= \int^\infty_{-\infty} e^{it\lambda}\Phi(d\lambda).NEWLINE\]NEWLINE The closed linear span of \(X(t)\), \(t\in{\mathbf R}\), under the Schilder's norm \(({\mathcal A},\|\cdot\|)_\alpha)\) is the time domain of the process which is a Banach space of jointly S\(\alpha\)S random variables. If \({\mathcal S}_0\) is the linear span of the elements of the set \(\{X(t), t\in{\mathbf R}\}\), a scalar product \(\langle\cdot,\cdot\rangle\) is defined. It is proved that the completion of \(({\mathcal S}_0,\langle\cdot, \cdot\rangle)\) is a Hilbert space, \(({\mathcal S},\langle\cdot,\cdot\rangle_{\mathcal S})\), of jointly symmetric stable random variables with certain properties. Furtheron a moving average representation against a stable random measure \(Z\) exists in \({\mathcal S}\), and \(Z\) has orthogonal increments in \(({\mathcal S},\langle\cdot, \cdot\rangle_{{\mathcal S}})\). The best linear predictor of \(X(t+ T)\) based on \(\{X_s, s\leq t\}\) is given in spectral form and in moving average form.