9,438 research outputs found
Weak differentiability of product measures
In this paper, we study cost functions over a finite collection of random variables. For these types of models, a calculus of differentiation is developed that allows us to obtain a closed-form expression for derivatives where "differentiation" has to be understood in the weak sense. The technique for proving the results is new and establishes an interesting link between functional analysis and gradient estimation. The key contribution of this paper is a product rule of weak differentiation. In addition, a product rule of weak analyticity is presented that allows for Taylor series approximations of finite products measures. In particular, from characteristics of the individual probability measures, a lower bound (i.e., domain of convergence) can be established for the set of parameter values for which the Taylor series converges to the true value. Applications of our theory to the ruin problem from insurance mathematics and to stochastic activity networks arising in project evaluation review techniques are provided. © 2010 INFORMS
The Lip-lip equality is stable under blow-up
We show that at generic points blow-ups/tangents of differentiability spaces
are still differentiability spaces; this implies that an analytic condition
introduced by Keith as an inequality (and later proved to actually be an
equality) passes to tangents. As an application, we characterize the -weak
gradient on iterated blow-ups of differentiability spaces.Comment: minor corrections: change of normalization of the measures; The final
version will appear in Calc. Var. PD
Functional delta-method for the bootstrap of quasi-Hadamard differentiable functionals
The functional delta-method provides a convenient tool for deriving the
asymptotic distribution of a plug-in estimator of a statistical functional from
the asymptotic distribution of the respective empirical process. Moreover, it
provides a tool to derive bootstrap consistency for plug-in estimators from
bootstrap consistency of empirical processes. It has recently been shown that
the range of applications of the functional delta-method for the asymptotic
distribution can be considerably enlarged by employing the notion of
quasi-Hadamard differentiability. Here we show in a general setting that this
enlargement carries over to the bootstrap. That is, for quasi-Hadamard
differentiable functionals bootstrap consistency of the plug-in estimator
follows from bootstrap consistency of the respective empirical process. This
enlargement often requires convergence in distribution of the bootstrapped
empirical process w.r.t.\ a nonuniform sup-norm. The latter is not problematic
as will be illustrated by means of examples
Critical -differentiability of -maps and canceling operators
We give a generalization of Dorronsoro's Theorem on critical
-Taylor expansions for -maps on ,
i.e., we characterize homogeneous linear differential operators of
-th order such that has -th order
-Taylor expansion a.e. for all
(here , with an
appropriate convention if ). The space
consists of those locally integrable maps
such that is a Radon measure on . A new
-Sobolev inequality is established to cover higher order
expansions. Lorentz refinements are also considered. The main results can be
seen as pointwise regularity statements for linear elliptic systems with
measure-data.Comment: 29 pages; to appear in Transactions of the American Mathematical
Societ
On the volume measure of non-smooth spaces with Ricci curvature bounded below
We prove that, given an -space , then it is possible
to -essentially cover by measurable subsets
with the following property: for each there exists such that is absolutely continuous
with respect to the -dimensional Hausdorff measure. We also show that a
Lipschitz differentiability space which is bi-Lipschitz embeddable into a
euclidean space is rectifiable as a metric measure space, and we conclude with
an application to Alexandrov spaces.Comment: Final version to appear in the Annali della Scuola Normale Superiore
Classe di Scienz
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