Exchangeability and related topics

The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space X to have the property that bounded additive X-valued maps on o-algebras be strongly bounded are presented, namely, X can contain no copy of /„. The next two sections treat the Jordan decomposition for measures with values in Z.|-spaces on c0(r) spaces and criteria for integrability of scalar functions with respect to vector measures. In particular, a different proof is presented for a result of D. R. Lewis to the effect that scalar integrability implies integrability is equivalent to noncontainment of c0. The final section concerns the Radon-Nikodym theorem for vector measures. A generalization of a result due to E. Leonard and K. Sundaresan is given, namely, if a Banach space X has an equivalent very smooth norm (in particular, a Fréchet differentiable normithenitsdualhas the Radon-Nikodym property. Consequently, a C(H) space which is a Grothendieck space (weak-star and weak-sequential convergence in dual coincide) cannot be renormed smoothly. Several open questions are mentioned throughout the paper. The present paper contains results on various aspects of the general theory of vector-valued measures. It proceeds in four sections which are unrelated to each other except for their general relationship to the topic of the title. A brief outline of the results of each section is presented below—a more complete discussion of the sections is delayed (largely because of their disconnected nature) until the sections themselves. §1 is concerned with the theory of strongly bounded vector measures. The main result of this section (Theorem 1.1) provides criteria for a Banach space X to possess the property that every X-valued bounded additive map with values in X be strongly bounded. This theorem sharpens the classical Pettis theorem on weakly countably additive set functions and allows a sharpening of several other related results. §2 is concerned with the Jordan decomposition of vector measures with values in a Banach lattice. The results of this section are necessarily meager: not much is possible. Our most precise results are in case the range space is an abstract £space or c0. A few remarks are also made concerning the range of certain vector measures. §3 deals with the integrability of certain scalar functions with respect to a vector measure. Utilizing the series representation of a scalar function and its integral, a result of D. R. Lewis is generalized. Also, a criterion for integrability Received by the editors February 5, 1973 and, in revised form, May 25, 1973. AMS (MOS) subject classifications (1970). Primary 46B05.

On vector measures

Recent years have witnessed an explosion in the volume and variety of data collected in all scientific disciplines and industrial settings. Such massive data sets present a number of challenges to researchers in statistics and machine learning. This book provides a self-contained introduction to the area of high-dimensional statistics, aimed at the first-year graduate level. It includes chapters that are focused on core methodology and theory - including tail bounds, concentration inequalities, uniform laws and empirical process, and random matrices - as well as chapters devoted to in-depth exploration of particular model classes - including sparse linear models, matrix models with rank constraints, graphical models, and various types of non-parametric models. With hundreds of worked examples and exercises, this text is intended both for courses and for self-study by graduate students and researchers in statistics, machine learning, and related fields who must understand, apply, and adapt modern statistical methods suited to large-scale data.

High-Dimensional Statistics: A Non-Asymptotic Viewpoint

Date: First version: June 2009, this version October 28, 2010. Preliminary results of this paper were FIRST presented at Chernozhukov's invited Cowles Foundation lecture at the Northern American meetings of the Econometric society in June of 2009. We thank seminar participants at Brown, Columbia, Harvard-MIT, the Dutch Econometric Study Group, Fuqua School of Business, and NYU for helpful comments. We also thank Denis Chetverikov, JB Doyle, and Joonhwan Lee for thorough reading of this paper and helpful feedback.

https://www.princeton.edu/~erp/erp%20seminar%20pdfs/papersFall2011/victor.pdf

Sparse models and methods for optimal instruments with an application to eminent domain

Given i.i.d. observations of a random vector X 2 R p , we study the problem of estimating both its covariance matrix � ∗ , and its inverse covariance or concentration matrix � ∗ = (� ∗ ) −1 . We estimate � ∗ by minimizing an l1-penalized log-determinant Bregman divergence; in the multivariate Gaussian case, this approach corresponds to l1-penalized maximum likelihood, and the structure of � ∗ is specified by the graph of an associated Gaussian Markov random field. We analyze the performance of this estim ator under high-dimensional scaling, in which the number of nodes in the graph p, the number of edges s and the maximum node degree d, are allowed to grow as a function of the sample size n. In addition to the parameters (p, s, d), our analysis identifies other key quantities that control rates: (a) the l∞-operator norm of the true covariance matrix � ∗ ; and (b) the l∞ operator norm of the submatrix ∗, where S indexes the graph edges, and ∗ = (� ∗ ) −1 (� ∗ ) −1 ; and (c) a mutual incoherence or irrepresentability measure on the matrix ∗ and (d) the rate of decay 1/f(n, δ) on the probabilities {|b n � ∗| > δ}, where b � n is the sample covariance based on n samples. Our first result establishes consistency of our estimate b � in the elementwise maximum-norm. This in turn allows us to derive convergence rates in Frobenius and spectral norms, with improvements upon existing results for graphs with maximum node degrees d = o( p s). In our second result, we show that with probability converging to one, the estimate b � correctly specifies the zero pattern of the concentration matrix � ∗ . We illustrate our theoretical results via simulations for various graphs and problem parameters, showing good correspondences between the theoretical predictions and behavior in simulations. 1. Introduction. The area of high-dimensional statistics deals with estimation in the “large p, small n” setting, where p and n correspond, respectively, to the dimensionality of the dat a and the sample size. Such high-dimensional problems arise in a variety of applications, among them remote sensing, computational biology and natural language processing, where the model dimension may be comparable or substantially larger than the sample size. It is well-known that such high-dimensional scaling can lead to dramatic breakdowns in many classical procedures. In the absence of additional model assumptions, it is frequently impossible to obtain consistent procedures when p ≫ n. Accordingly, an active line of statistical research is based on imposing various restrictions on the model—-for instance, sparsity, manifold structure, or graphical model structure—-and then studying the scaling behavior of different estimators as a function of sample size n, ambient dimension p and additional parameters related to these structural assu mptions.

/pdf/high-dimensional-covariance-estimation-by-minimizing-l1-183hiocmie.pdf

High-dimensional covariance estimation by minimizing ℓ1-penalized log-determinant divergence

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inL
 p
 , each of mean zero, is shown to be isomorphic tol
 2,l
 p
 ,l
 2⊕l
 p
 , or a new spaceX
 p
 , and the linear topological properties ofX
 p
 are investigated. It is proved thatX
 p
 is isomorphic to a complemented subspace ofL
 p
 and another uncomplemented subspace ofL
 p
 , whence there exists an uncomplemented subspace ofl
 p
 isomorphic tol
 p
 . It is also proved thatX
 p
 is not isomorphic to the previously knownℒ
 p
 spaces.

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separableℒ
 p
 space has a basis.

On bases, finite dimensional decompositions and weaker structures in Banach spaces

/pdf/on-relatively-disjoint-families-of-measures-with-some-lcsvggw325.pdf

Haskell P. Rosenthal

Papers

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

On bases, finite dimensional decompositions and weaker structures in Banach spaces

On relatively disjoint families of measures, with some applications to Banach space theory

On ω*-basic sequences and their applications to the study of Banach spaces

On Subspaces of L p