OscillatoryProcess - crowlogic/arb4j GitHub Wiki

A process $$ X_t $$ is called an oscillatory process if it has the representation $$ X_t = \int e^{it\lambda} a_t(\lambda) Z(d\lambda), \quad t \in \mathbb{R} $$ where $$ a_t(\lambda) $$ is a function depending on both $$ t $$ and $$ \lambda $$, and $$ Z(d\lambda) $$ is a random measure.

This definition generalizes the representation of harmonizable processes, allowing the function $$ a_t(\lambda) $$ to depend on time, which introduces the oscillatory, or evolving, nature of the process. The process is termed "oscillatory" because the function $$ a_t(\lambda) $$ can vary with $$ t $$, leading to nonstationary behavior, as opposed to stationary processes where $$ a_t(\lambda) $$ would be independent of $$ t $$

Certainly! Here is the rigorous mathematical definition of an oscillatory process in the context of harmonizable stochastic processes, as you described:

Rigorous Mathematical Definition

A stochastic process $$ (X_t){t\in\mathbb{R}} $$ is called an oscillatory process if it admits a representation of the form $$ X_t = \int{\mathbb{R}} e^{it\lambda}, a_t(\lambda), Z(d\lambda), \qquad t \in \mathbb{R}, $$ where

  • $$ a_t(\lambda) $$ is a deterministic (possibly complex-valued) function depending on both time $$ t \in \mathbb{R} $$ and frequency $$ \lambda \in \mathbb{R} $$.
  • $$ Z(d\lambda) $$ is a (complex-valued) orthogonal random measure on $$ \mathbb{R} $$ with control measure $$ \mu $$.

Additional Comments

  • The function $$ a_t(\lambda) $$ is measurable and chosen so that the stochastic integral is well-defined (e.g., the map $$ t \mapsto a_t(\lambda) $$ is measurable for each $$ \lambda $$, and for each $$ t $$, the function $$ \lambda \mapsto a_t(\lambda) $$ is square-integrable with respect to the control measure).
  • The oscillatory nature arises because $$ a_t(\lambda) $$ may depend non-trivially on $$ t $$, leading to nonstationary and potentially highly varying behavior over time.
  • If $$ a_t(\lambda) $$ does not depend on $$ t $$, i.e. $$ a_t(\lambda) = a(\lambda) $$, then the process reduces to a (wide sense) stationary harmonizable process.

Reference

  • I.A. Ibragimov, Y.A. Rozanov, Gaussian Random Processes, Section 3.2.4 (Oscillatory processes).
  • B.B. Mandelbrot, "Harmonizable Processes," Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (1965).

Summary:
An oscillatory process generalizes harmonizable processes by allowing the coefficient function $$ a_t(\lambda) $$ in the spectral representation to depend on time, thereby capturing nonstationary and evolving behaviors. The defining integral representation is $$ X_t = \int e^{it\lambda} a_t(\lambda) Z(d\lambda). $$ with the terms as defined above.