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Journal ArticleDOI

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.
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TL;DR: In this paper, the authors extend the classical notion of the spherical depth in \mathbb{R}^k, to the important setup of data on a Riemannian manifold, and show that this notion of depth satisfies a set of desirable properties.
Abstract: We extend the classical notion of the spherical depth in \mathbb{R}^k, to the important setup of data on a Riemannian manifold. We show that this notion of depth satisfies a set of desirable properties. For the empirical version of this depth function both uniform consistency and the asymptotic distribution are studied. Consistency is also shown for functional data. The behaviour of the depth is illustrated through several examples for Riemannian manifold data.
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TL;DR: In this article, the authors established Chung's law of the iterated logarithm for a class of anisotropic Gaussian random fields with stationary increments, which is applicable to space-time Gaussian Random Fields and solution to the stochastic fractional heat equation.
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Journal ArticleDOI
TL;DR: In this article, the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed $p$-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for pure $p $-spin models and mixed perturbations of them, was derived.
Abstract: We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed $p$-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure $p$-spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature).
Journal ArticleDOI
TL;DR: In this paper , the authors proposed two novel schemes of wideband compressive spectrum sensing via block orthogonal matching pursuit (BOMP) algorithm, for achieving high sensing accuracy in real time.
Abstract: This paper proposes two novel schemes of wideband compressive spectrum sensing (CSS) via block orthogonal matching pursuit (BOMP) algorithm, for achieving high sensing accuracy in real time. These schemes aim to reliably recover the spectrum by adaptively adjusting the number of required measurements without inducing unnecessary sampling redundancy. To this end, the minimum number of required measurements for successful recovery is first derived in terms of its probabilistic lower bound. Then, a CSS scheme is proposed by tightening the derived lower bound, where the key is the design of a nonlinear exponential indicator through a general-purpose sampling-controlled algorithm (SCA). In particular, a sampling-controlled BOMP (SC-BOMP) is developed through a holistic integration of the existing BOMP and the proposed SCA. For fast implementation, a modified version of SC-BOMP is further developed by exploring the block orthogonality in the form of sub-coherence of measurement matrices, which allows more compressive sampling in terms of smaller lower bound of the number of measurements. Such a fast SC-BOMP scheme achieves a desired tradeoff between the complexity and the performance. Simulations demonstrate that the two SC-BOMP schemes outperform the other benchmark algorithms.
References
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Book
01 Jan 1953

10,512 citations

Journal ArticleDOI
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)....

    [...]

BookDOI
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

Journal ArticleDOI
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....

    [...]