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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Dissertation•
10 Dec 2009
TL;DR: In this paper, a trois modeles classiques de geometrie aleatoire : les mosaiques, les enveloppes convexes and le modele booleen.
Abstract: Ce memoire est consacre a trois modeles classiques de geometrie aleatoire : les mosaiques, les enveloppes convexes et le modele booleen. Dans la premiere partie, on etudie les mosaiques poissonniennes d'hyperplans isotropes et plus particulierement leur zero-cellule qui est un polyedre convexe aleatoire de l'espace euclidien. Deux cas particuliers de zero-cellules sont la cellule typique de Poisson-Voronoi et la cellule de Crofton. On donne une formule explicite pour la loi du nombre de cotes d'une zero-cellule en dimension deux. On s'interesse au comportement asymptotique de cette loi et on fait le lien avec le probleme de Sylvester des points en position convexe. On decrit ensuite la loi du rayon circonscrit ainsi que le comportement asymptotique du polyedre a grand rayon inscrit au moyen de theoremes limites. De cette maniere et aussi par l'utilisation de la frequence fondamentale, on apporte des precisions a l'enonce de la conjecture de D. G. Kendall. La seconde partie a pour objet les enveloppes convexes de processus ponctuels de Poisson isotropes dans la boule-unite. On etablit un resultat de type grandes deviations pour le nombre de sommets. On montre ensuite la convergence de la frontiere de l'enveloppe apres changement d'echelle et on en deduit des resultats de valeurs extremes, estimations de variance, theoremes centraux limites et principes d'invariance pour certaines caracteristiques. Dans la troisieme partie, on s'interesse enfin aux modeles de recouvrement de type booleen de l'espace euclidien. Dans un premier travail, on applique une variante du modele sans interpenetration des objets a la modelisation d'un phenomene de fissuration. On etudie ensuite la convergence de la composante connexe de l'origine d'un modele booleen vers la cellule de Crofton en dimension deux. On s'interesse enfin a la fonction de visibilite de cette composante connexe pour laquelle on obtient une estimee de la queue de distribution et des resultats de valeurs extremes.
4 citations
TL;DR: In this paper, a transformation of a discrete-time martingale with conditionally Gaussian increments into a sequence of i.i.d. standard Gaussian random variables is proposed as based on the sequence of stopping times constructed using the quadratic variation.
Abstract: A transformation of a discrete-time martingale with conditionally Gaussian increments into a sequence of i.i.d. standard Gaussian random variables is proposed as based on a sequence of stopping times constructed using the quadratic variation. It is shown that sequential estimators for the parameters in AR(1) and generalized first-order autoregressive models have a nonasymptotic normal distribution.
4 citations
TL;DR: In this article, a nonparametric model for the analysis of the two-treatment, two-period, four-sequence crossover design was proposed and tests for the equality of treatment effects and for the absence of carryover effects were provided.
Abstract: The paper describes a nonparametric model for the analysis of the two-treatment, two-period, four-sequence crossover design. Tests for the equality of treatment effects and for the absence of carryover effects are provided. Some relevant asymptotic results of the proposed tests are obtained and compared with the existing nonparametric and parametric competitors via simulation. It is shown that the proposed tests are comparable to or better than the competitors in terms of type I error and power.
4 citations
Posted Content•
TL;DR: A survey of quasi-sure results via the connection to the Brownian sheet can be found in this article, where it is shown that quasi-every continuous function satisfies the local law of the iterated logarithm, has Levy's modulus of continuity for Brownian motion, and is nowhere differentiable.
Abstract: We present a self-contained and modern survey of some existing quasi-sure results via the connection to the Brownian sheet. Among other things, we prove that quasi-every continuous function: (i) satisfies the local law of the iterated logarithm; (ii) has Levy's modulus of continuity for Brownian motion; (iii) is nowhere differentiable; and (iv) has a nontrivial quadratic variation. We also present a hint of how to extend (iii) to obtain a quasi-sure refinement of the M. Csorgo--P. Revesz modulus of continuity for almost every continuous function along the lines suggested by M. Fukushima.
4 citations
Cites background from "The integral of a symmetric unimoda..."
...14) P { ∃(s, t) ∈ [1, 2](2) : |W (s, t) − x| ≤ ε } ≤ 2CdKd(n+ 1)(2) [ ε + n ] ....
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...Let us fix ε ∈ (0, 1) and an integer n ≥ 1, and consider the covering [1, 2](2) = ∪i,j=0Ii,j , where Ii,j := [1 + (i/n), 1 + (i+ 1)/n] × [1 + (j/n), 1 + (j + 1)/n]....
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...In particular, if d ≥ 5, then with probability one, x 6∈W ([1, 2](2)), as claimed....
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...15) sup x∈R P { ∃(s, t) ∈ [1, 2](2) : |W (s, t) − x| ≤ ε } = O ( ε ) , (ε→ 0)....
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TL;DR: In this article, the authors derived a large deviation theorem for the distribution of the end-point of a one-dimensional diffusion with drift, where the drift function is a drift function and the real number is a real number.
Abstract: We derive a large deviation principle which describes the behaviour of a diffusion process with additive noise under the influence of a strong drift. Our main result is a large deviation theorem for the distribution of the end-point of a one-dimensional diffusion with drift $\theta b$ where $b$ is a drift function and $\theta$ a real number, when $\theta$ converges to $\infty$. It transpires that the problem is governed by a rate function which consists of two parts: one contribution comes from the Freidlin-Wentzell theorem whereas a second term reflects the cost for a Brownian motion to stay near a equilibrium point of the drift over long periods of time.
4 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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