On interpolation problem for multidimensional harmonizable stable sequences with noise observations

TL;DR

This paper develops optimal linear estimation methods for multidimensional harmonizable stable sequences with noise, focusing on the interpolation problem for unknown sequence values based on noisy observations.

Contribution

It introduces a new approach for optimal linear estimation of vector-valued harmonizable stable sequences with noise, extending previous work to multidimensional cases with spectral density conditions.

Findings

Derived explicit formulas for optimal estimators.

Established conditions for the minimality of spectral measures.

Extended interpolation theory to multidimensional stable sequences.

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.