26,960 research outputs found
Stein's method for comparison of univariate distributions
We propose a new general version of Stein's method for univariate
distributions. In particular we propose a canonical definition of the Stein
operator of a probability distribution {which is based on a linear difference
or differential-type operator}. The resulting Stein identity highlights the
unifying theme behind the literature on Stein's method (both for continuous and
discrete distributions). Viewing the Stein operator as an operator acting on
pairs of functions, we provide an extensive toolkit for distributional
comparisons. Several abstract approximation theorems are provided. Our approach
is illustrated for comparison of several pairs of distributions : normal vs
normal, sums of independent Rademacher vs normal, normal vs Student, and
maximum of random variables vs exponential, Frechet and Gumbel.Comment: 41 page
Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups
In the present paper, we develop geometric analytic techniques on Cayley
graphs of finitely generated abelian groups to study the polynomial growth
harmonic functions. We develop a geometric analytic proof of the classical
Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic
functions on lattices \mathds{Z}^n that does not use a representation formula
for harmonic functions. We also calculate the precise dimension of the space of
polynomial growth harmonic functions on finitely generated abelian groups.
While the Cayley graph not only depends on the abelian group, but also on the
choice of a generating set, we find that this dimension depends only on the
group itself.Comment: 15 pages, to appear in Ann. Global Anal. Geo
Positivity, entanglement entropy, and minimal surfaces
The path integral representation for the Renyi entanglement entropies of
integer index n implies these information measures define operator correlation
functions in QFT. We analyze whether the limit , corresponding
to the entanglement entropy, can also be represented in terms of a path
integral with insertions on the region's boundary, at first order in .
This conjecture has been used in the literature in several occasions, and
specially in an attempt to prove the Ryu-Takayanagi holographic entanglement
entropy formula. We show it leads to conditional positivity of the entropy
correlation matrices, which is equivalent to an infinite series of polynomial
inequalities for the entropies in QFT or the areas of minimal surfaces
representing the entanglement entropy in the AdS-CFT context. We check these
inequalities in several examples. No counterexample is found in the few known
exact results for the entanglement entropy in QFT. The inequalities are also
remarkable satisfied for several classes of minimal surfaces but we find
counterexamples corresponding to more complicated geometries. We develop some
analytic tools to test the inequalities, and as a byproduct, we show that
positivity for the correlation functions is a local property when supplemented
with analyticity. We also review general aspects of positivity for large N
theories and Wilson loops in AdS-CFT.Comment: 36 pages, 10 figures. Changes in presentation and discussion of
Wilson loops. Conclusions regarding entanglement entropy unchange
Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions
In this paper, we present a minimal formalism for Stein operators which leads
to different probabilistic representations of solutions to Stein equations.
These in turn provide a wide family of Stein-Covariance identities which we put
to use for revisiting the very classical topic of bounding the variance of
functionals of random variables. Applying the Cauchy-Schwarz inequality yields
first order upper and lower Klaassen-type variance bounds. A probabilistic
representation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder)
leads to Papathanasiou-type variance expansions of arbitrary order. A matrix
Cauchy-Schwarz inequality leads to Olkin-Shepp type covariance bounds. All
results hold for univariate target distribution under very weak assumptions (in
particular they hold for continuous and discrete distributions alike). Many
concrete illustrations are provided
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