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Prediction problem for continuous time stochastic processes with periodically correlated increments observed with noise

Maksym Luz, Mikhail Moklyachuk

TL;DR

The paper addresses mean-square optimal prediction of linear functionals of a continuous-time process with periodically correlated $d$-th increments observed in noise with periodic stationarity. It maps the non-stationary problem to an infinite-dimensional vector stationary-sequence framework and uses Hilbert-space projection to derive optimal linear estimates and their mean-square errors under spectral certainty. For spectral uncertainty, it develops a minimax-robust approach, characterizing least-favorable densities and minimax spectral characteristics across several admissible density classes via constrained optimization. The methodology extends classical stationary-process prediction to cyclostationary increment processes and provides computable formulas for robust prediction in signal-processing settings with periodic structure.

Abstract

We propose solution of the problem of the mean square optimal estimation of linear functionals which depend on the unobserved values of a continuous time stochastic process with periodically correlated increments based on observations of this process with periodically stationary noise. To solve the problem, we transform the processes to the sequences of stochastic functions which form an infinite dimensional vector stationary sequences. In the case of known spectral densities of these sequences, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

Prediction problem for continuous time stochastic processes with periodically correlated increments observed with noise

TL;DR

The paper addresses mean-square optimal prediction of linear functionals of a continuous-time process with periodically correlated -th increments observed in noise with periodic stationarity. It maps the non-stationary problem to an infinite-dimensional vector stationary-sequence framework and uses Hilbert-space projection to derive optimal linear estimates and their mean-square errors under spectral certainty. For spectral uncertainty, it develops a minimax-robust approach, characterizing least-favorable densities and minimax spectral characteristics across several admissible density classes via constrained optimization. The methodology extends classical stationary-process prediction to cyclostationary increment processes and provides computable formulas for robust prediction in signal-processing settings with periodic structure.

Abstract

We propose solution of the problem of the mean square optimal estimation of linear functionals which depend on the unobserved values of a continuous time stochastic process with periodically correlated increments based on observations of this process with periodically stationary noise. To solve the problem, we transform the processes to the sequences of stochastic functions which form an infinite dimensional vector stationary sequences. In the case of known spectral densities of these sequences, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.
Paper Structure (7 sections, 22 theorems, 198 equations)

This paper contains 7 sections, 22 theorems, 198 equations.

Table of Contents

  1. Continuous time periodically correlated processes and generated vector stationary sequences
  2. Stochastic processes with periodically correlated $d$th increments
  3. Hilbert space projection method of prediction
  4. Prediction based on factorizations of spectral densities
  5. Minimax (robust) method of prediction
  6. Least favorable spectral densities in classes $\mathcal{D}_0 \times \mathcal{D}_{1\delta}$
  7. Least favorable spectral densities in classes $\mathcal{D}_{\varepsilon}\times \mathcal{D}_V^U$

Key Result

Theorem 2.1

A stationary increment sequence $\xi^{(d)}_{j}$ is uniquely represented in the form where $\xi^{(d)}_{R,kj}, k=1,\ldots,\infty$, is a regular stationary increment sequence and $\xi^{(d)}_{S,kj}, k=1,\dots,\infty$, is a singular stationary increment sequence. The increment sequences $\xi^{(d)}_{R,kj}$ and $\xi^{(d)}_{S,kj}$ are orthogonal for all $j\in\mathbb{Z}$. They are defined b

Theorems & Definitions (33)

  • Definition 1.1: Gladyshev Glad1963
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1: Luz_Mokl_extra_cont_PCI
  • Lemma 3.2
  • proof
  • Remark 3.1
  • ...and 23 more