
StoneWeierstrass type theorems for large deviations

Henri Comman, University of Santiago de Chile 
Abstract
We give a general version of Bryc's theorem valid on any topological
space and with any algebra $mathcal{A}$ of realvalued continuous
functions separating the points, or any wellseparating class. In
absence of exponential tightness, and when the underlying space is
locally compact regular and $mathcal{A}$ constituted by functions
vanishing at infinity, we give a sufficient condition on the
functional $Lambda(cdot)_{mid mathcal{A}}$ to get large
deviations with not necessarily tight rate function. We obtain the
general variational form of any rate function on a completely
regular space; when either exponential tightness holds or the space
is locally compact Hausdorff, we get it in terms of any algebra as
above. Prohorovtype theorems are generalized to any space, and
when it is locally compact regular the exponential tightness
can be replaced by a (strictly weaker) condition on
$Lambda(cdot)_{mid mathcal{A}}$.

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Published on: April 28, 2008

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