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Orthogonal series for si- and related processes, Karhunen-Loève decompositions

Kacha Dzhaparidze

Abstract

This paper reproduces results from Chapter 11 of the forthcoming book \cite{dzh25}. It discusses series expansions of processes with stationary increments (si-processes) and certain associated processes. Making use of de Branges theory of Hilbert spaces of entire functions, it sheds new light on the existing literature and makes available some new results. In particular, it provides some new decompositions of the Karhunen-Loève type.

Orthogonal series for si- and related processes, Karhunen-Loève decompositions

Abstract

This paper reproduces results from Chapter 11 of the forthcoming book \cite{dzh25}. It discusses series expansions of processes with stationary increments (si-processes) and certain associated processes. Making use of de Branges theory of Hilbert spaces of entire functions, it sheds new light on the existing literature and makes available some new results. In particular, it provides some new decompositions of the Karhunen-Loève type.

Paper Structure

This paper contains 35 sections, 33 theorems, 347 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. De Branges spaces
  4. Theorem \ref{['thm:thm40p']}
  5. Theorem \ref{['thm:thm44p']}
  6. Isometry
  7. Moving average w.r.t. fundamental martingales
  8. Fundamental martingales
  9. Moving average
  10. Paley-Wiener series
  11. Sampling formula
  12. PW-series
  13. Remark.
  14. Proof of \ref{['eq:pawlisp']}.
  15. PW-series for FBM
  16. ...and 20 more sections

Key Result

Theorem 2.1

(i) Given an arbitrary spectral measure $\mu$ of property eq:sqipaper, there exists a short de Branges space $\mathcal{H}(E)$ of exponential type which is contained isometrically in $L^2(\mu)$.

Theorems & Definitions (58)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • Example 2.7
  • Definition 2.8
  • Theorem 2.9
  • Theorem 3.1
  • ...and 48 more