Abstract
In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance $$ \int^{s\wedge t}_0 u^a [(t-u)^b+(s-u)^b]du, $$ parameters $a>-1$, $-1 < b\leq 1$, $|b|\leq 1+a$, corresponds to fractional Brownian motion for $a=0$, $-1 < b < 1$. The second one, with covariance $$ (2-h)\biggl(s^h+t^h-\frac{1}{2}[(s+t)^h +|s-t|^h]\biggr), $$ parameter $0 < h\leq 4$, corresponds to sub-fractional Brownian motion for $0 < h < 2 $. The third one, with covariance $$ -\left(s^2\log s + t^2\log t -\frac{1}{2}[(s+t)^2 \log (s+t) +(s-t)^2 \log |s-t|]\right), $$ is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.
Citation
Tomasz Bojdecki. Luis Gorostiza. Anna Talarczyk. "Some Extensions of Fractional Brownian Motion and Sub-Fractional Brownian Motion Related to Particle Systems." Electron. Commun. Probab. 12 161 - 172, 2007. https://doi.org/10.1214/ECP.v12-1272
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