| Summary: | 碩士 === 國立成功大學 === 數學系應用數學碩博士班 === 96 === A well-known Stein's theorem asserts that, in order that a
real-valued random variable $mathbf{X}$ has a standard normal
distribution it is necessary and sufficient that, for all
continuous and piecewise continuously differentiable functions
$f:mathbb{R}
ightarrowmathbb{R}$ with finite $Emid
f'(mathbf{X})mid$, the following identity holds:
egin{eqnarray}label{a1}
E[f'(mathbf{X})]=E[mathbf{X}f(mathbf{X})].
end{eqnarray}
In this paper we concern ourself with the generalization of
Stein's characterization to a random variable $mathbf{X}$ taking
values in a real separable Banach space $B$. Our main result show
that, in order that the probability measure
$mu=Pcircmathbf{X}^{-1}$ of $mathbf{X}$ is standard Gaussian
it is necessary and sufficient that, there exists a real separable
Hilbert space $H$ such that $(i,H,B)$ is an abstract Wiener space
in the sense of Gross and the following identity holds:
egin{eqnarray}label{a2}
int_{B}(x,z)f((x,y))mu(dx) = l z,ygint_{B}f'((x,y))mu(dx),
end{eqnarray}
for $y, zin B^{*}$and for any differentiable function $f$ such that
$int_{B}|f'((x,y))|mu(dx)$ is finite, where $(cdot,cdot)$ and
$lcdot,cdotg$ denote respectively the $B-B^*$ pairing and the
inner product of $H$.
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