On the maximum of a fractional Brownian motion
Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 111-115
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Let $b_{\gamma}(t)$, $b_{\gamma}(0)=0$ be a fractional Brownian motion, i.e., a Gaussian process with the structural function $\mathbf{E}|b_{\gamma}(t)-b_{\gamma}(s)|^2=|t-s|^\gamma$, $0 \gamma 2$. The logarithmic asymptotics as $T\to\infty$ is found for the probabilities $P_T=\mathsf{P}\{b_{\gamma}(t)1,\ -\rho T0\}$ this asymptotics is independent of $\gamma$.
Keywords:
extreme values, Gaussian processes, fractional Brownian motion, automodel processes.
@article{TVP_1999_44_1_a7,
author = {G. M. Molchan},
title = {On the maximum of a fractional {Brownian} motion},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {111--115},
publisher = {mathdoc},
volume = {44},
number = {1},
year = {1999},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a7/}
}
G. M. Molchan. On the maximum of a fractional Brownian motion. Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 111-115. http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a7/