 00195 G.M. Molchan
 Maximum of fractional Brownian motion: probabilities of small values
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Apr 21, 00

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Abstract. Let $b_\gamma (t)$, $b_\gamma(0)=0$ be fractional Brownian
motion, i.e., a Gaussian process with the structure
function $Eb_\gamma (t)  b_\gamma (s)^2 = ts^\gamma$, $0<\gamma<2$.
We study the logarithmic asymptotics of
$P_T = P\{b_\gamma (t) < 1,\quad t \in T\Delta \}$ as
$T \to \infty$, where $\Delta$ is either the interval $(0,1)$
or a bounded region that contains a vicinity of $0$ for the case of
multidimensional time. It is shown that
$\log\,P_T = D\log\,T(1+o(1))$, where $D$ is the dimension of zeroes
of $b_\gamma(t)$ in the former case and the dimension of time in
the latter.
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