## Electronic Communications in Probability

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

Alexandre Richard

#### 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
Accepted: 26 February 2016
First available in Project Euclid: 6 April 2016

https://projecteuclid.org/euclid.ecp/1459966700

Digital Object Identifier
doi:10.1214/16-ECP4727

Mathematical Reviews number (MathSciNet)
MR3485400

Zentralblatt MATH identifier
1336.60073

#### 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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