Open Access
2016 Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization
Alexandre Richard
Electron. Commun. Probab. 21: 1-15 (2016). DOI: 10.1214/16-ECP4727


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.


Download Citation

Alexandre Richard. "Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization." Electron. Commun. Probab. 21 1 - 15, 2016.


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

zbMATH: 1336.60073
MathSciNet: MR3485400
Digital Object Identifier: 10.1214/16-ECP4727

Primary: 28C20 , 60G10 , 60G15 , 60G57 , 60G60

Keywords: fractional Brownian motion , Random fields , ‎spectral representation , stationarity

Back to Top