On the asymptotic form of convex hulls of Gaussian random fields
Tóm tắt
We consider a centered Gaussian random field X = {X
t : t ∈ T} with values in a Banach space
$$\mathbb{B}$$
defined on a parametric set T equal to ℝ
m
or ℤ
m
. It is supposed that the distribution
of X
t is independent of t. We consider the asymptotic behavior of closed convex hulls W
n
= conv{X
t : t ∈ T
n}, where (T
n
) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b
n
)
n≥1 with probability 1,
(in the sense of Hausdorff distance), where the limit set
is the concentration ellipsoid of
. The asymptotic behavior of the mathematical expectations Ef(W
n
), where f is some function, is also studied.
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