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

Alexandre Richard (1)

1 TOSCA - TO Simulate and CAlibrate stochastic models

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.

Mots clés: Random fields; Stationarity; Spectral representation; Fractional Brownian motion

Domaines: Probabilités [math.PR]

Alexandre Richard. Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization. Electronic Communications in Probability, 2016, 21, pp.15. ⟨10.1214/16-ECP4727⟩. ⟨hal-01236156v2⟩