Steiner symmetrization using a finite set of directions

Let $v_1, ..., v_m$ be a finite set of unit vectors in \RR^n. Suppose that an infinite sequence of Steiner symmetrizations are applied to a compact convex set $K$ in \RR^n, where each of the symmetrizations is taken with respect to a direction from among the $v_i$. Then the resulting sequence of Steiner symmetrals always converges, and the limiting body is symmetric under reflection in any of the directions $v_i$ that appear infinitely often in the sequence. In particular, an infinite periodic sequence of Steiner symmetrizations always converges, and the set functional determined by this infinite process is always idempotent.Comment: 18 pages. The essential results are the same as in the previous version. This version includes a more thorough introduction, some clarifications in the proofs, an updated bibliography, and some open questions at the en

Towards Conformal Invariance and a Geometric Representation of the 2D Ising Magnetization Field

We study the continuum scaling limit of the critical Ising magnetization in two dimensions. We prove the existence of subsequential limits, discuss connections with the scaling limit of critical FK clusters, and describe work in progress of the author with C. Garban and C.M. Newman.Comment: 20 pages, 1 figure, presented at the workshop "Inhomogeneous Random Systems" held at IHP (Paris) on January 26-27, 201

Randomly Weighted Self-normalized L\'evy Processes

Let $(U_t,V_t)$ be a bivariate L\'evy process, where $V_t$ is a subordinator and $U_t$ is a L\'evy process formed by randomly weighting each jump of $V_t$ by an independent random variable $X_t$ having cdf $F$. We investigate the asymptotic distribution of the self-normalized L\'evy process $U_t/V_t$ at 0 and at $\infty$. We show that all subsequential limits of this ratio at 0 ($\infty$) are continuous for any nondegenerate $F$ with finite expectation if and only if $V_t$ belongs to the centered Feller class at 0 ($\infty$). We also characterize when $U_t/V_t$ has a non-degenerate limit distribution at 0 and $\infty$.Comment: 32 page

Planar Ising magnetization field I. Uniqueness of the critical scaling limit

The aim of this paper is to prove the following result. Consider the critical Ising model on the rescaled grid $a\mathbb{Z}^2$, then the renormalized magnetization field $$\Phi^a:=a^{15/8}\sum_{x\in a\mathbb{Z}^2}\sigma_x\delta_x,$$ seen as a random distribution (i.e., generalized function) on the plane, has a unique scaling limit as the mesh size $a\searrow0$. The limiting field is conformally covariant.Comment: Published in at http://dx.doi.org/10.1214/13-AOP881 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Necessary conditions for variational regularization schemes

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization method, we start with an investigation of the converse question: How could necessary conditions for a variational method to provide a regularization method look like? To this end, we formalize the notion of a variational scheme and start with comparison of three different instances of variational methods. Then we focus on the data space model and investigate the role and interplay of the topological structure, the convergence notion and the discrepancy functional. Especially, we deduce necessary conditions for the discrepancy functional to fulfill usual continuity assumptions. The results are applied to discrepancy functionals given by Bregman distances and especially to the Kullback-Leibler divergence.Comment: To appear in Inverse Problem

Upward and downward statistical continuities

A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset $E$ of $\textbf{R}$, is statistically upward compact if any sequence of points in $E$ has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in $E$ has a statistically downward half quasi-Cauchy subsequence where a sequence $(x_{n})$ of points in $\textbf{R}$ is called statistically upward half quasi-Cauchy if $$\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k}-x_{k+1}\geq \varepsilon\}|=0$$ is statistically downward half quasi-Cauchy if $$\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+1}-x_{k}\geq \varepsilon\}|=0$$ for every $\varepsilon>0$. We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that uniform limit of a sequence of statistically upward continuous functions is statistically upward continuous, and uniform limit of a sequence of statistically downward continuous functions is statistically downward continuous.Comment: 25 pages. arXiv admin note: substantial text overlap with arXiv:1205.3674, arXiv:1103.1230, arXiv:1102.1531, arXiv:1305.069