Electronic Communications in Probability

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

Alexandre Richard

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Abstract

We are interested in the increment stationarity property of $L^2$-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 Ito [9], 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.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 31, 15 pp.

Dates
Received: 20 November 2015
Accepted: 26 February 2016
First available in Project Euclid: 6 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1459966700

Digital Object Identifier
doi:10.1214/16-ECP4727

Mathematical Reviews number (MathSciNet)
MR3485400

Zentralblatt MATH identifier
1336.60073

Subjects
Primary: 60G10: Stationary processes 60G57: Random measures 60G60: Random fields 60G15: Gaussian processes 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
stationarity random fields spectral representation fractional Brownian motion

Rights
Creative Commons Attribution 4.0 International License.

Citation

Richard, Alexandre. Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization. Electron. Commun. Probab. 21 (2016), paper no. 31, 15 pp. doi:10.1214/16-ECP4727. https://projecteuclid.org/euclid.ecp/1459966700


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