Maximum of a Fractional Brownian Motion: Probabilities of Small Values |
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Authors: | G M Molchan |
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Institution: | (1) International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Warshavskoye sh. 79, kor. 2, Moscow, 113556, Russia. E-mail: molchan@mitp.ru, RU |
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Abstract: | Let b
γ (t), b
γ(0)= 0 be a fractional Brownian motion, i.e., a Gaussian process with the structure function , 0 < γ < 2. We study the logarithmic asymptotics of P
T
= P{b
γ (t) < 1,□t∈TΔ} as T→∞, where Δ 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 ln P
T
= - D ln T(1 + o(1)), where D is the dimension of zeroes of b
γ (t) in the former case and the dimension of time in the latter.
Received: 28 September 1998 / Accepted: 19 February 1999 |
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Keywords: | |
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