Diagonally canonical and related Gaussian random elements
V. V. Kvaratskhelia,
V. I. Tarieladze Muskhelishvili Institute of Computational Mathematics
Abstract:
We call a Gaussian random element
$\eta$ in a Banach space
$X$ with a Schauder basis
$\mathbf{e}=(e_n)$ diagonally canonical (for short,
$D$-canonical) with respect to
$\mathbf{e}$ if the distribution of
$\eta$ coincides with the distribution of a random element having the form
$B\xi$, where
$\xi$ is a Gaussian random element in
$X$, whose
$\mathbf{e}$-components are stochastically independent and
$B:X\to X$ is a continuous linear mapping. In this paper we show that if
$X=l_p$,
$1\leqq p<\infty$ and
$p\ne2$, or
$X=c_0$, then there exists a Gaussian random element
$\eta$ in
$X$, which is not
$D$-canonical with respect to the natural basis of
$X$. We derive this result in the case when
$X=l_p$,
$2<p<\infty$, or
$X=c_0$ from the following statement, an analogue which was known earlier only for Banach spaces without an unconditional Schauder basis: if
$X=l_p$,
$2<p<\infty$, or
$X=c_0$, then there exists a Gaussian random element
$\eta$ in
$X$ such that the distribution of
$\eta$ does not coincide with the distribution of the sum of almost surely convergent in
$X$ series
$\sum_{n=1}^\infty x_ng_n$, where
$(x_n)$ is an unconditionally summable sequence of elements of
$X$ and
$(g_n)$ is a sequence of stochastically independent standard Gaussian random variables.
Keywords:
diagonally canonical gaussian random element; unconditionally canonical gaussian random element; gaussian covariance operator; cotype of Banach spaces; r-nuclear operator; summing operator; Gaussian average property; $gl_2$-Banach space.
MSC: 60 Received: 31.08.2011
Revised: 01.10.2012
DOI:
10.4213/tvp4507
Fulltext:
PDF file (182 kB)