Robust extrapolation problem for random processes with stationary increments

TL;DR

This work tackles extrapolation of linear functionals $Aξ$ and $A_Tξ$ for random processes with stationary $n$th increments using past observations. It develops a Hilbert-space projection approach and spectral representation to yield mean-square-optimal linear estimators and spectral characteristics under spectral certainty, with explicit formulas for the associated error. It then extends to spectral uncertainty via a minimax-robust framework, deriving least-favorable spectral densities and robust spectral characteristics for classes $\\mathcal{D}_0$, $\\\mathcal{D}_v^u$, and $\\\mathcal{D}_{δ}$ through conditional extremum (saddle-point) problems. The results provide a principled method for robust extrapolation in time series and signal processing when spectral information is incomplete, using canonical factorization, innovation representations, and one-sided moving-average forms to characterize and compute the estimators.

Abstract

The problem of optimal estimation of linear functionals $A ξ=\int_{0}^{\infty} a(t)ξ(t)dt$ and $A_Tξ=\int_{0}^{T} a(t)ξ(t)dt$ depending on the unknown values of random process $ξ(t)$, $t\in R$, with stationary $n$th increments from observations of ttis process for $t<0$ is considered. Formulas for calculating mean square error and spectral characteristic of optimal linear estimation of the functionals are proposed in the case when spectral density is exactly known. Formulas that determine the least favorable spectral densities are proposed for given sets of admissible spectral densities.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Remark 1