On interpolation problem for multidimensional harmonizable stable sequences with noise observations
Mikhail Moklyachuk
TL;DR
The paper tackles optimal linear interpolation of a linear functional $A_N\vec{\xi}$ of a vector-valued harmonizable symmetric $α$-stable sequence from observations contaminated by noise. It develops a projection-based framework in the $α$-stable setting, deriving the spectral characteristic $h(\theta)$ and the mean-square error under spectral certainty, and then extends to spectral uncertainty via minimaxRobust analysis. For uncertain spectra, it identifies least favorable densities and minimax spectral characteristics across several natural classes, including fixed-energy, moment-constrained, and inverse-density classes, using Lagrange multipliers and convex-analytic tools. The results yield robust, closed-form prescriptions for estimating $A_N\vec{\xi}$ in the presence of noise and spectral misspecification, with explicit expressions in several important special cases and connections to autoregressive representations. Overall, the work provides a principled, robust interpolation framework for HS$α$S sequences with practical relevance to signal processing under heavy-tailed noise and spectral uncertainty.
Abstract
We consider the problem of optimal linear estimation of the functional $$A_N \vecξ =\sum_{j = 0}^{N} (\vec{a}(j))^{\top} \vecξ(j)$$ that depends on the unknown values $\vecξ(j),j=0,1,\dots,N,$ of a vector-valued harmonizable symmetric $α$-stable random sequence $\vecξ(j)=\left \{ ξ_ {k} (j) \right \}_{k = 1} ^ {T}$, from observations of the sequence $\vecξ(j)+\vecη(j)$ at points $j\in\mathbb Z\setminus\{0,1,\dots,N\}$. We consider the problem for mutually independent vector-valued harmonizable symmetric $α$-stable random sequences $\vecξ(j)=\left \{ ξ_ {k} (j) \right \}_{k = 1} ^ {T}$ and $\vecη(j)=\left \{ ξ_ {k} (j) \right \}_{k = 1} ^ {T}$ which have absolutely continuous spectral measures and the spectral densities $f(θ)$ and $g(θ)$ satisfying the minimality condition.