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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.
Citations
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Journal ArticleDOI
Manabu Iwasa1
TL;DR: In this article, the symmetry condition of G-monotone functions is weakened by orderings obtained from a cone, and Bartholomew's conjectures about directions yielding the maximum power and minimum power of likelihood ratio tests for order-restricted alternatives are partly settled.

1 citations

Posted Content
TL;DR: In this paper, it was shown that for any physically implementable family of oscillator-to-oscillator codes with a constant amount of squeezing and maximum likelihood error decoding, a lower bound on the logical error probability is only a function of the number of modes.
Abstract: It is known that continuous variable quantum information cannot be protected against naturally occurring noise using Gaussian states and operations only. Noh et al. (PRL 125:080503, 2020) proposed bosonic oscillator-to-oscillator codes relying on non-Gaussian resource states as an alternative, and showed that these encodings can lead to a reduction of the effective error strength at the logical level as measured by the variance of the classical displacement noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N-mode unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states. Here we ask if - in analogy to qubit error-correcting codes - there are families of oscillator-to-oscillator codes with the following threshold property: They allow to convert physical displacement noise with variance below some threshold value to logical noise with variance upper bounded by any (arbitrary) constant. We find that this is not the case if encoding unitaries involving a constant amount of squeezing and maximum likelihood error decoding are used. We show a general lower bound on the logical error probability which is only a function of the amount of squeezing and independent of the number of modes. As a consequence, any physically implementable family of oscillator-to-oscillator codes combined with maximum likelihood error decoding does not admit a threshold.

1 citations

Journal ArticleDOI
TL;DR: In this article, the authors obtained the precise asymptoties of a kind of weighted infinite series for partial sums SN as ǫ ↘ 0 by using the truncation and approximation methods.
Abstract: Consider the positive d-dimensional lattice ℤ+d (d≥2) with partial ordering ≤, let {XK; K e ℤ+d} be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖·‖) with mean zero and covariance operator Σ, and set partial sums SN =ΣK≤NXK, K, N e ℤ+d. Under some moment conditions, we obtain the precise asymptoties of a kind of weighted infinite series for partial sums SN as ɛ ↘0 by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of (Gut and Spǎtaru, 2003).

1 citations


Cites background from "The integral of a symmetric unimoda..."

  • ...definitive, thus by applying Corollary 3 of ( Anderson, 1955 ), we have that for any x∈\ and ,...

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Journal ArticleDOI
TL;DR: In this paper, it was shown that a univariate probability distribution with support in [−1, 1] and unimodal with respect to 0 has a variance of at most 13, which is attained only for the uniform distribution.

1 citations

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....

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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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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....

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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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