Convex rearrangements of gaussian fields TeX Export

@article{citeulike:5758022, abstract = {Convexification, also called convex rearrangement, is an operation defined for absolutely continuous functions on a compact interval of \$\mathbb{R}\$. The so-called \emph{convexified} of such a function \$f\$ is the unique convex function defined on the same interval whose derivative distribution is the same as that of \$f'\$, and is obtained by rearranging the increments of \$f\$ in ascending order. When \$f\$ is irregular, the consensus is to investigate the asymptotic convexification of regularizations. By extension, we define here the convexified of a smooth multivariate function with compact convex support as the unique convex function having the same gradient distribution. In this article, we discuss the asymptotic convexification of regularizations of random fields, with focus on Gaussian fields on the compact \$[0,1]^d\$. Given an irregular random field \$X\$, we consider polygonal approximations interpolating \$X\$ in the vertices of a triangulation whose diameter goes to 0. We give general results concerning the eventual limit after renormalization, with stronger results of convergence for gaussian fields. Along with other examples, we use these to show the existence of a limit convexified of the \$d\$-dimensional Chentsov field for a wide class of triangulations, and give the limit gradient distribution. In the 2-dimensional case, we give a tractable expression and a graphical representation of the limit convexified for a natural triangulation.}, archivePrefix = {arXiv}, author = {Lachieze-Rey, Raphael and Davydov, Youri}, citeulike-article-id = {5758022}, citeulike-linkout-0 = {http://arxiv.org/abs/0909.1288}, citeulike-linkout-1 = {http://arxiv.org/pdf/0909.1288}, day = {7}, eprint = {0909.1288}, keywords = {convex, gaussian, random-field}, month = {Sep}, posted-at = {2009-09-09 12:32:27}, priority = {2}, title = {Convex rearrangements of gaussian fields}, url = {http://arxiv.org/abs/0909.1288}, year = {2009} }

TY - JOUR ID - citeulike:5758022 L3 - citeulike-article-id:5758022 N2 - Convexification, also called convex rearrangement, is an operation defined for absolutely continuous functions on a compact interval of $\mathbb{R}$. The so-called \emph{convexified} of such a function $f$ is the unique convex function defined on the same interval whose derivative distribution is the same as that of $f'$, and is obtained by rearranging the increments of $f$ in ascending order. When $f$ is irregular, the consensus is to investigate the asymptotic convexification of regularizations. By extension, we define here the convexified of a smooth multivariate function with compact convex support as the unique convex function having the same gradient distribution. In this article, we discuss the asymptotic convexification of regularizations of random fields, with focus on Gaussian fields on the compact $[0,1]^d$. Given an irregular random field $X$, we consider polygonal approximations interpolating $X$ in the vertices of a triangulation whose diameter goes to 0. We give general results concerning the eventual limit after renormalization, with stronger results of convergence for gaussian fields. Along with other examples, we use these to show the existence of a limit convexified of the $d$-dimensional Chentsov field for a wide class of triangulations, and give the limit gradient distribution. In the 2-dimensional case, we give a tractable expression and a graphical representation of the limit convexified for a natural triangulation. TI - Convex rearrangements of gaussian fields KW - convex KW - gaussian KW - random-field AU - Lachieze-Rey, Raphael AU - Davydov, Youri PY - 2009/Sep/07/ UR - http://arxiv.org/abs/0909.1288 ER -