On interpolation problem for multidimensional harmonizable stable sequences with noise observations

Abstract

We consider the problem of optimal linear estimation of the functional $$A_N\vec{\xi }=\sum _{j=0}^N(\vec{a}(j){)}^{\top }\vec{\xi }(j)$$ that depends on the unknown values $\vec{\xi }(j),j=0,1,\dots ,N,$ of a vector-valued harmonizable symmetric $\alpha$-stable random sequence $\vec{\xi }(j)={\left\{{\xi }_k(j)\right\}}_{k=1}^T$, from observations of the sequence $\vec{\xi }(j)+\vec{\eta }(j)$ at points $j\in \mathbb{Z}\setminus \{0,1,\dots ,N\}$. We consider the problem for mutually independent vector-valued harmonizable symmetric $\alpha$-stable random sequences $\vec{\xi }(j)={\left\{{\xi }_k(j)\right\}}_{k=1}^T$ and $\vec{\eta }(j)={\left\{{\xi }_k(j)\right\}}_{k=1}^T$ which have absolutely continuous spectral measures and the spectral densities $f(\theta )$ and $g(\theta )$ satisfying the minimality condition.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1607.00754, arXiv:2406.17917