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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01 Jan 1980TL;DR: In this paper, two fundamental inequalities with greater generality were studied, i.e., the probability content of a convex set that is symmetric about the origin, and a generalization of an inequality of Sidak (see Theorem 2.2.2).
Abstract: Two fundamental inequalities with greater generality will be studied in this chapter. Basically they involve the probability content of a convex set that is symmetric about the origin. The first inequality was obtained by Anderson (1955), and we have already seen a special case of it (Lemma 2.2.2) when applied to multivariate normal distribution. Generalizations of this inequality will be discussed in Section 4.2. The second inequality, due to Das Gupta, Eaton, Olkin, Perlman, Savage, and Sobel (1972), represents a generalization of an inequality of Sidak (see Theorem 2.2.5) and yields a number of new interesting results for the normal, t, chi-square, and F distributions.
TL;DR: In this paper , the authors studied the probability that a real stationary Gaussian process has at least ηT zeros in [0, T ] (overcrowding), or at most this number (undercrowded) and showed that if the spectral measure of the process is supported on ± [B, A ], overcrowding probability transitions from exponential decay to Gaussian decay at η = Aπ , while undercrowdness probability undergoes the reverse transition at δ = Bπ .
Abstract: . We study the probability that a real stationary Gaussian process has at least ηT zeros in [0 , T ] (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on ± [ B, A ], overcrowding probability transitions from exponential decay to Gaussian decay at η = Aπ , while undercrowding probability undergoes the reverse transition at η = Bπ .
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TL;DR: In this paper , a fast iterative configuration method for MPLC was proposed, following a mathematical review of the matrix distance and proposal of a new norm, which showed that adding a few redundant layers significantly improves the convergence of the MPLC architecture, making it a practical and attractive option.
Abstract: Programmable unitary photonic devices are emerging as promising tools to implement unitary transformation for quantum information processing, machine learning, and optical communication. These devices typically use a rectangular mesh of Mach-Zehnder interferometers (MZIs), which has a clear mathematical structure and can be configured deterministically. However, this mesh architecture is sensitive to fabrication errors, and the correction techniques are still under investigation. In contrast, the multi-plane light conversion (MPLC) architecture is more robust against fabrication errors, but a deterministic method for configuring the converter has not yet been developed due to its complex mathematical structure. In this work, we propose a fast iterative configuration method for MPLC, following the mathematical review of the matrix distance and proposal of a new norm. We show through numerical simulations that adding a few redundant layers significantly improves the convergence of the MPLC architecture, making it a practical and attractive option. We also consider the effects of finite resolution and crosstalk in phase shifters in our simulations. In addition, we propose a phase-insensitive distance suited for applications using only intensity detections. Our method demonstrates orders of magnitude better accuracy and a 20-fold speed-up compared to previous approaches.
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TL;DR: In this paper , the existence of an exact uniform modulus continuity for the class of q-isotropic Gaussian random processes was proved for a d-dimensional version of the B γ Gaussian processes.
Abstract: We find sufficient conditions for the existence of an exact uniform modulus continuity for the class of q -isotropic Gaussian random fields introduced in [8]. We apply the result to a d -dimensional version of the B γ Gaussian processes defined in [14].
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01 Jan 1981TL;DR: In this paper, the authors studied the monotonicity properties of distributions in order to obtain inequalities useful in statistical inference, and showed that some of these properties are well known and have proved to be very useful.
Abstract: It is very important to study the monotonicity properties of distributions in order to obtain inequalities useful in statistical inference. Some monotonicity properties of distributions are well known and have proved to be very useful. During the last decade, more concepts have been introduced and used by several authors in multiple decision problems.
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
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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
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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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