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.