EXTREME VALUES OF DERIVATIVES OF SMOOTHED FRACTIONALBROWNIAN MOTIONS
Sándor Csörgő Jan Mielniczuk
Abstract: Let be a fractional Brownian motion on with parameter
and consider its smoothed version
where the kernel is a density function and the are some bandwidths. The
derivative of this process arises naturally as a heuristic approximation of a nonparametric
kernel regression estimator when the normal errors are long-range dependent. We show that,
with suitable centering and norming, the distribution of the supremum and absolute
supremum of this derivative over the interval converges, as to the
Gumbel extreme-value distribution and its square, respectively. A version of the
problem for finite differences is also considered, along with higher-order derivatives.
be a fractional Brownian motion on 
with parameter
and consider its smoothed version 
where the kernel 
is a density function and the 
are some bandwidths. The
derivative of this process arises naturally as a heuristic approximation of a nonparametric
kernel regression estimator when the normal errors are long-range dependent. We show that,
with suitable centering and norming, the distribution of the supremum and absolute
supremum of this derivative over the interval ![[0,1]](files/16.2/HTML/16.2.3.abs8x.png)
converges, as 
to the
Gumbel extreme-value distribution and its square, respectively. A version of the
problem for finite differences is also considered, along with higher-order derivatives.