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
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TL;DR: In this paper, sample path properties of isotropic spherical Gaussian fields on S 2 have been investigated based on the high-frequency behavior of its angular power spectrum, and the authors exploit this result to establish an exact uniform modulus of continuity for its sample paths.
20 citations
TL;DR: This paper investigates a multiuser MIMO relay downlink system with imperfect channel estimation and limited feedback, and presents a limited feedback strategy for both two-hop channels in the relay system to guarantee a bounded rate loss for growing SNRs.
Abstract: This paper investigates a multiuser MIMO relay downlink system with imperfect channel estimation and limited feedback. We analyze the achievable rate loss due to channel state information (CSI) mismatch arising from both channel estimation and quantization feedback. We first derive an upper bound to characterize the effect of imperfect CSI, and then present a limited feedback strategy for both the two-hop channels in the relay system. This newly presented feedback strategy reveals the relationship among the CSI quantization levels, the transmit power, and the pilots for channel estimation. An optimized pilot design at the base station and the relay is also presented by applying the derived bounds for the two-hop system.
20 citations
Cites methods from "The integral of a symmetric unimoda..."
...This can be proved by using Corollary 3 in [12]....
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TL;DR: A novel linear discriminant analysis approach for the classification of high-dimensional matrix-valued data that commonly arises from imaging studies is proposed using an efficient nuclear norm penalized regression that encourages a low-rank structure.
Abstract: We propose a novel linear discriminant analysis (LDA) approach for the classification of high-dimensional matrix-valued data that commonly arises from imaging studies. Motivated by the equivalence ...
20 citations
Cites background from "The integral of a symmetric unimoda..."
...(Anderson’s comparison inequality (Anderson 1955)) Let X and Y be zero-mean Gaussian random vectors with covariance X and Y, respectively....
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TL;DR: In this article, it was shown that if the probability measure is positive semidefinite, then it is more peaked about zero than a probability measure that is positive definite, in the sense of Sherman (Ann. Math. Statist., 26 (1955), pp. 763-766).
Abstract: Let $\mu _\Sigma $ and $\mu _\Omega $ be probability measures on $\mathcal{R}^n $ symmetric about zero with matrices ${\boldsymbol \Sigma }$ and ${\boldsymbol \Omega }$ as scale parameters. If ${\boldsymbol \Omega} - {\boldsymbol \Sigma} $ is positive semidefinite, then for suitable measures $\mu _\Sigma $ is more peaked about zero than $\mu _\Omega $ in the sense of Sherman (Ann. Math. Statist., 26 (1955), pp. 763–766). This was shown for Gaussian measures by Anderson (Proc. Amer. Math. Soc., 6 (1955), pp. 170–176), and for ellipsoidal measures by Das Gupta et al. (Sixth Berkeley Symposium on Probability and Statistics, II (1971), pp. 241–265) and Fefferman et al. (Proc. Amer. Math. Soc., 33 (1972), pp. 114–119), when $\Omega $ and $\Sigma $ are positive definite. Here these results are extended to include singular measures and the converse is proved. Marginal and conditional versions of ordered measures are shown to be ordered, and some applications are noted.
20 citations
TL;DR: Nane et al. as mentioned in this paper derived general results on the small deviation behavior for some classes of iterated processes, and gave a new and correct proof of some results in E. Nane, Electron. J. Probab. 11 (2006), no. 18, 434--459.
Abstract: We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for n-iterated Brownian motions and, more generally, for the iteration of n fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Electron. J. Probab. 11 (2006), no. 18, 434--459.
20 citations
References
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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....
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...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)....
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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....
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...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....
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