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Preservation of log-concavity on summation

Oliver Johnson, Christina Goldschmidt (2006)

ESAIM: Probability and Statistics

We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of...

Probability distribution solutions of a general linear equation of infinite order

Tomasz Kochanek, Janusz Morawiec (2009)

Annales Polonici Mathematici

Let (Ω,,P) be a probability space and let τ: ℝ × Ω → ℝ be strictly increasing and continuous with respect to the first variable, and -measurable with respect to the second variable. We obtain a partial characterization and a uniqueness-type result for solutions of the general linear equation F ( x ) = ∫ Ω F ( τ ( x , ω ) ) P ( d ω ) in the class of probability distribution functions.

Probability distribution solutions of a general linear equation of infinite order, II

Tomasz Kochanek, Janusz Morawiec (2010)

Annales Polonici Mathematici

Let (Ω,,P) be a probability space and let τ: ℝ × Ω → ℝ be a mapping strictly increasing and continuous with respect to the first variable, and -measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation F ( x ) = ∫ Ω F ( τ ( x , ω ) ) P ( d ω ) . We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.

Properties of the induced semigroup of an Archimedean copula

Włodzimierz Wysocki (2004)

Applicationes Mathematicae

It is shown that to every Archimedean copula H there corresponds a one-parameter semigroup of transformations of the interval [0,1]. If the elements of the semigroup are diffeomorphisms, then it determines a special function v H called the vector generator. Its knowledge permits finding a pseudoinverse y = h(x) of the additive generator of the Archimedean copula H by solving the differential equation d y / d x = v H ( y ) / x with initial condition ( d h / d x ) ( 0 ) = - 1 . Weak convergence of Archimedean copulas is characterized in terms of vector...

Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ?

Laurent Miclo (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

Classically, Hardy’s inequality enables to estimate the spectral gap of a one-dimensional diffusion up to a factor belonging to [ 1 , 4 ] . The goal of this paper is to better understand the latter factor, at least in a symmetric setting. In particular, we will give an asymptotical criterion implying that its value is exactly 4. The underlying argument is based on a semi-explicit functional for the spectral gap, which is monotone in some rearrangement of the data. To find it will resort to some regularity...

Quantile of a Mixture with Application to Model Risk Assessment

Carole Bernard, Steven Vanduffel (2015)

Dependence Modeling

We provide an explicit expression for the quantile of a mixture of two random variables. The result is useful for finding bounds on the Value-at-Risk of risky portfolios when only partial dependence information is available. This paper complements the work of [4].

Currently displaying 761 – 780 of 1158