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On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

Djalil Chafaï, Florent Malrieu (2010)

Annales de l'I.H.P. Probabilités et statistiques

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed...

On functional measures of skewness

Renata Dziubińska, Dominik Szynal (1996)

Applicationes Mathematicae

We introduce a concept of functional measures of skewness which can be used in a wider context than some classical measures of asymmetry. The Hotelling and Solomons theorem is generalized.

On Gaussian Brunn-Minkowski inequalities

Franck Barthe, Nolwen Huet (2009)

Studia Mathematica

We are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrhard inequality for m Borel or convex sets based on a previous work by Borell. Our method also yields semigroup proofs of the geometric Brascamp-Lieb inequality and of its reverse form, which follow exactly the same lines.

On generalized conditional cumulative past inaccuracy measure

Amit Ghosh, Chanchal Kundu (2018)

Applications of Mathematics

The notion of cumulative past inaccuracy (CPI) measure has recently been proposed in the literature as a generalization of cumulative past entropy (CPE) in univariate as well as bivariate setup. In this paper, we introduce the notion of CPI of order α and study the proposed measure for conditionally specified models of two components failed at different time instants, called generalized conditional CPI (GCCPI). Several properties, including the effect of monotone transformation and bounds of GCCPI...

On lower bounds for the variance of functions of random variables

Faranak Goodarzi, Mohammad Amini, Gholam Reza Mohtashami Borzadaran (2021)

Applications of Mathematics

In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable X , we find a characterization of the weighted function corresponding to density function f ( x ) , in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual entropy...

On Measure Concentration of Vector-Valued Maps

Michel Ledoux, Krzysztof Oleszkiewicz (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in k . To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.

On preservation under univariate weighted distributions

Salman Izadkhah, Mohammad Amini, Gholam Reza Mohtashami Borzadaran (2015)

Applications of Mathematics

We derive some new results for preservation of various stochastic orders and aging classes under weighted distributions. The corresponding reversed preservation properties as straightforward conclusions of the obtained results for the direct preservation properties, are developed. Damage model of Rao, residual lifetime distribution, proportional hazards and proportional reversed hazards models are discussed as special weighted distributions to try some of our results.

On reliability analysis of consecutive k -out-of- n systems with arbitrarily dependent components

Ebrahim Salehi (2016)

Applications of Mathematics

In this paper, we consider the linear and circular consecutive k -out-of- n systems consisting of arbitrarily dependent components. Under the condition that at least n - r + 1 components ( r n ) of the system are working at time t , we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k -out-of- n systems. In the following, we investigate the inactivity time of the component with lifetime T r : n at the system...

On Talagrand's deviation inequalities for product measures

Michel Ledoux (2010)

ESAIM: Probability and Statistics

We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincaré and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with theses tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced...

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