Source: Adv. in Appl. Probab. Volume 37, Number 3
(2005), 743-764.
We study stationary processes given as solutions to stochastic differential equations
driven by fractional Brownian motion. This rich class includes
the fractional Ornstein-Uhlenbeck process and those processes that
can be obtained from it by state space
transformations. An explicit formula in terms of
Euler's Γ-function describes the asymptotic behaviour of
the covariance function of the fractional Ornstein-Uhlenbeck process
near zero, which, by an
application of Berman's condition, guarantees that this process is in
the maximum domain of attraction of the Gumbel distribution.
Necessary and sufficient conditions on the state space transforms
are stated to classify the maximum domain of attraction of
solutions to stochastic differential equations driven by fractional Brownian motion.
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