A Nonsymmetric Correlation Inequality for Gaussian Measure

In this paper we show that the family P d (lc) of probability distributions on ℝ d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. In this and several other respects the nonparametric model P d (lc) behaves like a parametric model such as, for instance, the family of all d-variate Gaussian distributions. As a consequence of the continuity result, we prove the existence of nontrivial confidence sets for the moments of an unknown distribution in P d (lc). Our results are based on various new inequalities for log-concave distributions which are of independent interest.

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“…[10] p. 17), while the inequality on the right is an improvement of an earlier result of Komatsu due to [23].…”

Section: Inequalities For φmentioning

confidence: 66%

“…Now instead of the Mills' ratio bound (16) for the tail of the normal distribution, we use the upper bound part of (18) due to [23]. This yields…”

Section: Inequalities For φmentioning

confidence: 99%

Multivariate log-concave distributions as a nearly parametric model

Schuhmacher¹,

Huesler

Duembgen

2011

Statistics &Amp; Risk Modeling

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“…The next non-symmetric result is related to a statement of Szarek & Werner [20]. They proved that, for every convex body A of R n and for every strip B symmetric with respect to the Gaussian barycenter of A, γ n (A∩B) γ n (A)γ n (B).…”

Section: Applications Of a Results Of Caffarellimentioning

confidence: 87%

Some Applications of Mass Transport to Gaussian-Type Inequalities

Cordero–Erausquin¹

2002

Archive for Rational Mechanics and Analysis

109

132

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As discovered by Brenier, mapping through a convex gradient gives the optimal transport in R n . In the present article, this map is used in the setting of Gaussianlike measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.

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“…In fact, a log-concave function is also quasi-concave (G()~x + (1 -~. )y) > min(G (x), G (y))) and so a proof of (5.2) for symmetric quasi-concave functions will also prove (6.1) (again see Pitt [16], and also Hu [7], Schechtman et al [21], and Szarek and Wermer [22]). Second order characterizations of smooth quasi-concave exist (see Crouzeix and Ferland [3]).…”

Section: Some Concluding Remarks On a Gaussian Correlation Conjecturementioning

confidence: 95%