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

Alexandre Richard 1, * 
* Corresponding author
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
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 Ito [7], 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.
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Submitted on : Friday, December 2, 2016 - 10:35:27 AM
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Alexandre Richard. Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2016, 21, pp.15. ⟨10.1214/16-ECP4727⟩. ⟨hal-01236156v2⟩



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