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## Propagation of Singularities in the Semi-Fractional Brownian Sheet

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Author(s) : Jochen Blath , Andreas Martin

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 25-2006

MSC 2000

60G15 Gaussian processes
60G17 Sample path properties
60G60 Random fields

Abstract :
Let $X$ be a semi-fractional Brownian sheet, that is a centred and continuous Gaussian random field with $\mathbb E [X(s,t)X(\hat{s},\hat{t}\,)] = (t\wedge \hat{t}\,) (s^\alpha + \hat{s}^\alpha-|s-\hat{s}|^\alpha)/2$. We provide, for $\alpha\in(0,2)$, an analysis of the propagation of singularities into the fractional direction of $X$. Here, singularities are times where the law of the iterated logarithm fails, such as fast points.

Keywords : Propagation of singularities, fractional Brownian motion, semi-fractional Brownian sheet, Gaussian random field, fast points