Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization

TL;DR

This paper develops a spectral representation for $L^2$-indexed stochastic processes and characterizes the $L^2$-indexed fractional Brownian motion through increment stationarity and self-similarity.

Contribution

It introduces a spectral representation theorem for $L^2$-indexed processes and characterizes the $L^2$-indexed fractional Brownian motion using increment stationarity and self-similarity.

Findings

Spectral representation theorem for $L^2$-indexed processes

Characterization of $L^2$-indexed fractional Brownian motion

Applications to random fields

Abstract

We are interested in the increment stationarity property for $L^2$-indexed stochastic processes, which is a fairly general concern since many random fields can be interpreted as the restriction of a more generally defined $L^2$-indexed process. We first give a spectral representation theorem in the sense of \citet{Ito54}, and see potential applications on random fields, in particular on the $L^2$-indexed extension of the fractional Brownian motion. Then we prove that this latter process is characterized by its increment stationarity and self-similarity properties, as in the one-dimensional case.