The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
01 Feb 1955-Vol. 6, Iss: 2, pp 170-176
About: The article was published on 1955-02-01 and is currently open access. It has received 552 citations till now. The article focuses on the topics: Convex set & Subderivative.
Citations
More filters
TL;DR: In this article, it was shown that the class of Banach spaces isomorphic to a Hilbert space under operations of passing to closed subspaces with a Schauder basis is closed.
Abstract: Introduction. Questions and problems related to Schauder basis are among the most classical ones in the Banach space theory and have been studied from the earliest days. The fact that every infinite-dimensional Banach space has an infinite-dimensional subspace with a basis, has been observed already in Banach's book [Ba]. At about the same time, the existence of bases in classical Banach spaces, such as C(0,1) and Lp(0,1), was established. An importance of the concept of a basis lies in the fact that it provides a natural method of approximation of vectors and operators in the space. A question related to the approximation property in Banach spaces was asked by Mazur in The Scottish Book [Scb]. This question and the problem of the existence of a basis in every Banach space stayed open for about 40 years, although not because of the lack of interest (cf. e.g. [Sin]). In 1972 P Enflo constructed a Banach space which fails the approximation property and hence does not admit a Schauder basis. His discovery triggered a flurry of activities resulting in constructions of subspaces failing the approximation property in all classical Banach spaces not isomorphic to a Hilbert space. Such subspaces have been also found in all Banach spaces sufficiently "far" from the Hilbert space (cf. e.g. [L-T.l], [L-T.2] and references therein). Techniques developed until that moment did not provide quantitative finitedimensional estimates for basis constants. It took another 10 years before a new probabilistic argument was introduced by E. D. Gluskin [G.l], and consequently the existence of n-dimensional Banach spaces with basis constant tending to infinity was proved independently by Gluskin [G.2] and S. J. Szarek [Sz.l]. At last a circle got closed: a method of constructing infinite-dimensional examples out of finite-dimensional ones was proposed by J. Bourgain [B.l] and developed by Szarek [Sz.3], to obtain an /2-sum of finite-dimensional Banach spaces which has no basis, but obviously admits a so-called finite-dimensional decomposition. In this paper we prove a striking isomorphic characterization of a Hilbert space in terms of a Schauder basis. First observe that the class of Banach spaces isomorphic to a Hilbert space is closed under operations of passing to closed
8 citations
TL;DR: A solution for the Gaussian version of the Busemann-Petty problem with additional information about dilates and translations is given and the size of theGaussian measure of the hyperplane sections of the dilates of the unit cube is discussed.
8 citations
Cites methods from "The integral of a symmetric unimoda..."
...We will also use one dimensional version of Anderson’s inequality for Gaussian measure (see [1] or [4, p....
[...]
TL;DR: In this article, the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager-Machlup functional of the posterior of the prior.
Abstract: We consider the inverse problem of recovering an unknown functional parameter $u$ in a separable Banach space, from a noisy observation $y$ of its image through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$ is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community.
Our key result is to show that in this non-parametric setup the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager--Machlup functional of the posterior. This is done independently for the so-called weak and strong MAP estimates, which as we show coincide in our context. In addition, we prove a form of weak consistency for the MAP estimators in the infinitely informative data limit. Our results are remarkable for two reasons: first, the prior distribution is non-Gaussian and does not meet the smoothness conditions required in previous research on non-parametric MAP estimates. Second, the result analytically justifies existing uses of the MAP estimate in finite but high dimensional discretizations of Bayesian inverse problems with the considered Besov priors.
8 citations
01 Jan 1984
TL;DR: Several inequalities and monotonicity results have been obtained in the study of selection and ranking problems; these, in fact, are germane to the development of the theory as mentioned in this paper, and the assumption regarding some order relations such as stochastic ordering and the monotone likelihood ratio property.
Abstract: Several inequalities and monotonicity results have been obtained in the study of selection and ranking problems; these, in fact, are germane to the development of the theory. Basic to the setup of these problems is the assumption regarding some order relations such as stochastic ordering and the monotone likelihood ratio property. These and other related ideas, along with some basic inequalities that arise under these assumptions are reviewed. Further, some important inequalities relevant to selection from restricted families of distributions defined by some partial order relations (such as IFR and IFRA families) are also discussed. Several specific results relating to multivariate normal, multinomial and gamma distributions are also reviewed.
8 citations
Cites result from "The integral of a symmetric unimoda..."
...7 (Anderson, 1955)....
[...]
...We will mention here only two results, namely, those of Anderson (1955) and Slepian (1962)....
[...]
TL;DR: In this paper, a Gaussian measureμ onE is a probability measure on (E,C) such that the law of eachf∈E′ is gaussian.
Abstract: LetE be a locally convex vector space,E′ its dual. ProvideE with the smallestσ-algebraC making measurable the elements ofE′. A gaussian measureμ onE is a probability measure on (E,C) such that the law of eachf∈E′ is gaussian. This setting is very general, but the use of measure-theoretic tools allows us to get many precise results (some of them being new even for Wiener measure). IfF denotes the algebraic dual ofE′, we define in the usual way the centerb μ ∈F ofμ and the reproducing kernelR μ ⊂E ofE.
8 citations
References
More filters
TL;DR: In this article, a general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations.
Abstract: The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty
3,082 citations
"The integral of a symmetric unimoda..." refers background in this paper
...In Theorem 1 the equality in (1) holds for k<l if and only if, for every u, (E+y)r\Ku=Er\Ku-\-y....
[...]
...It will be noticed that we obtain strict inequality in (1) if and only if for at least one u, H(u)>H*(u) (because H(u) is continuous on the left)....
[...]
1,660 citations
01 Jan 1934
TL;DR: In this article, Minkowski et al. den engen Zusammenhang dieser Begriffbildungen und Satze mit der Frage nach der bestimmung konvexer Flachen durch ihre GAusssche Krtim mung aufgedeckt und tiefliegende diesbeztigliche Satze bewiesen.
Abstract: Konvexe Figuren haben von jeher in der Geometrie eine bedeutende Rolle gespielt. Die durch ihre KonvexiUitseigenschaft allein charakteri sierten Gebilde hat aber erst BRUNN zum Gegenstand umfassender geometrischer Untersuchungen gemacht. In zwei Arbeiten "Ovale und EifHichen" und "Kurven ohne Wendepunkte" aus den Jahren 1887 und 1889 (vgl. Literaturverzeichnis BRUNN [1J, [2J) hat er neben zahl reichen Satzen der verschiedensten Art tiber konvexe Bereiche und Korper einen Satz tiber die Flacheninhalte von parallelen ebenen Schnitten eines konvexen K6rpers bewiesen, der sich in der Folge als fundamental herausgestellt hat. Die Bedeutung dieses Satzes hervor gehoben zu haben, ist das Verdienst von MINKOWSKI. In mehreren Arbeiten, insbesondere in "Volumeri. und Oberflache" (1903) und in der groBztigig angelegten, unvollendet geblieben n Arbeit "Zur Theorie der konvexen K6rper" (Literaturverzeichnis [3], [4J) hat er durch Ein fUhrung von grundlegenden Begriffen wie Stutzfunktion, gemischtes VolulIl, en usw. die dem Problemkreis angemessenen formalen Hilfsmittel geschaffen und vor allem den Weg zu vielseitigen Anwendungen, speziell auf das isoperimetrische (isepiphane) und andere Extremalprobleme fUr konvexe Bereiche und K6rper er6ffnet. Weiterhin hat MINKOWSKI den engen Zusammenhang dieser Begriffsbildungen und Satze mit der Frage nach der Bestimmung konvexer Flachen durch ihre GAusssche Krtim mung aufgedeckt und tiefliegende diesbeztigliche Satze bewiesen.
927 citations
TL;DR: In this paper, the authors extended the Cramer-Smirnov and von Mises test to the parametric case, a suggestion of Cramer [1], see also [2].
Abstract: The "goodness of fit" problem, consisting of comparing the empirical and hypothetical cumulative distribution functions (cdf's), is treated here for the case when an auxiliary parameter is to be estimated. This extends the Cramer-Smirnov and von Mises test to the parametric case, a suggestion of Cramer [1], see also [2]. The characteristic function of the limiting distribution of the test function is found by consideration of a Guassian stochastic process.
140 citations
"The integral of a symmetric unimoda..." refers background in this paper
...f ud[H*(u) - H(u)} = b[H*(b) - H(b)] - a[H*(a) - H(a)} (3) " + f [(H(u) - H*(u)]du....
[...]
...J a Since/(x) has a finite integral over E, bH(b)—>0 as b—>oo and hence also bH*(b)—>0 as b—*<x>; therefore the first term on the right in (3) can be made arbitrarily small in absolute value....
[...]