On the contraction method with degenerate limit equation

A class of random recursive sequences (Y_n) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X\stackrel{L}{=}X. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the ``unique'' solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000017

On the computation of Wasserstein barycenters

The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. In this paper, we state a general version of the equivalence of the Wasserstein barycenter problem to the n-coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n-coupling problem) provides a novel criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution the simple to implement iterative swapping algorithm (ISA) for computing barycenters. The ISA is a completely non-parametric algorithm which provides a sharp image of the support of the barycenter and has a quadratic time complexity which is comparable to other well established algorithms designed to compute barycenters. The algorithm can also be applied to more complex optimization problems like the k-barycenter problem

The nonlinear Bernstein-Schr\"odinger equation in Economics

In this paper we relate the Equilibrium Assignment Problem (EAP), which is underlying in several economics models, to a system of nonlinear equations that we call the "nonlinear Bernstein-Schr\"odinger system", which is well-known in the linear case, but whose nonlinear extension does not seem to have been studied. We apply this connection to derive an existence result for the EAP, and an efficient computational method.Comment: 8 pages, submitted to Lecture Notes in Computer Scienc

Mutations in pericentrin cause Seckel syndrome with defective ATR-dependent DNA damage signaling

Large brain size is one of the defining characteristics of modern humans. Seckel syndrome (MIM 210600), a disorder of markedly reduced brain and body size, is associated with defective ATR-dependent DNA damage signaling. Only a single hypomorphic mutation of ATR has been identified in this genetically heterogeneous condition. We now report that mutations in the gene encoding pericentrin (PCNT)--resulting in the loss of pericentrin from the centrosome, where it has key functions anchoring both structural and regulatory proteins--also cause Seckel syndrome. Furthermore, we find that cells of individuals with Seckel syndrome due to mutations in PCNT (PCNT-Seckel) have defects in ATR-dependent checkpoint signaling, providing the first evidence linking a structural centrosomal protein with DNA damage signaling. These findings also suggest that other known microcephaly genes implicated in either DNA repair responses or centrosomal function may act in common developmental pathways determining human brain and body size

A COL17A1 Splice-Altering Mutation Is Prevalent in Inherited Recurrent Corneal Erosions

PurposeCorneal dystrophies are a genetically heterogeneous group of disorders. We previously described a family with an autosomal dominant epithelial recurrent erosion dystrophy (ERED). We aimed to identify the underlying genetic cause of ERED in this family and 3 additional ERED families. We sought to characterize the potential function of the candidate genes using the human and zebrafish cornea.DesignCase series study of 4 white families with a similar ERED. An experimental study was performed on human and zebrafish tissue to examine the putative biological function of candidate genes.ParticipantsFour ERED families, including 28 affected and 17 unaffected individuals.MethodsHumanLinkage-12 arrays (Illumina, San Diego, CA) were used to genotype 17 family members. Next-generation exome sequencing was performed on an uncle–niece pair. Segregation of potential causative mutations was confirmed using Sanger sequencing. Protein expression was determined using immunohistochemistry in human and zebrafish cornea. Gene expression in zebrafish was assessed using whole-mount in situ hybridization. Morpholino-induced transient gene knockdown was performed in zebrafish embryos.Main Outcome MeasuresLinkage microarray, exome analysis, DNA sequence analysis, immunohistochemistry, in situ hybridization, and morpholino-induced genetic knockdown results.ResultsLinkage microarray analysis identified a candidate region on chromosome chr10:12,576,562–112,763,135, and exploration of exome sequencing data identified 8 putative pathogenic variants in this linkage region. Two variants segregated in 06NZ–TRB1 with ERED: COL17A1 c.3156C→T and DNAJC9 c.334G→A. The COL17A1 c.3156C→T variant segregated in all 4 ERED families. We showed biologically relevant expression of these proteins in human cornea. Both proteins are expressed in the cornea of zebrafish embryos and adults. Zebrafish lacking Col17a1a and Dnajc9 during development show no gross corneal phenotype.ConclusionsThe COL17A1 c.3156C→T variant is the likely causative mutation in our recurrent corneal erosion families, and its presence in 4 independent families suggests that it is prevalent in ERED. This same COL17A1 c.3156C→T variant recently was identified in a separate pedigree with ERED. Our study expands the phenotypic spectrum of COL17A1 disease from autosomal recessive epidermolysis bullosa to autosomal dominant ERED and identifies COL17A1 as a key protein in maintaining integrity of the corneal epithelium

Linkage to chromosome 2q32.2-q33.3 in familial serrated neoplasia (Jass syndrome)

Causative genetic variants have to date been identified for only a small proportion of familial colorectal cancer (CRC). While conditions such as Familial Adenomatous Polyposis and Lynch syndrome have well defined genetic causes, the search for variants underlying the remainder of familial CRC is plagued by genetic heterogeneity. The recent identification of families with a heritable predisposition to malignancies arising through the serrated pathway (familial serrated neoplasia or Jass syndrome) provides an opportunity to study a subset of familial CRC in which heterogeneity may be greatly reduced. A genome-wide linkage screen was performed on a large family displaying a dominantly-inherited predisposition to serrated neoplasia genotyped using the Affymetrix GeneChip Human Mapping 10 K SNP Array. Parametric and nonparametric analyses were performed and resulting regions of interest, as well as previously reported CRC susceptibility loci at 3q22, 7q31 and 9q22, were followed up by finemapping in 10 serrated neoplasia families. Genome-wide linkage analysis revealed regions of interest at 2p25.2-p25.1, 2q24.3-q37.1 and 8p21.2-q12.1. Finemapping linkage and haplotype analyses identified 2q32.2-q33.3 as the region most likely to harbour linkage, with heterogeneity logarithm of the odds (HLOD) 2.09 and nonparametric linkage (NPL) score 2.36 (P = 0.004). Five primary candidate genes (CFLAR, CASP10, CASP8, FZD7 and BMPR2) were sequenced and no segregating variants identified. There was no evidence of linkage to previously reported loci on chromosomes 3, 7 and 9

ON REGRESSION REPRESENTATIONS OF STOCHASTIC-PROCESSES

We construct a.s. nonlinear regression representations of general stochastic processes (X(n))n is-an-element-of N. As a consequence we obtain in particular special regression representations of Markov chains and of certain m-dependent sequences. For m-dependent sequences we obtain a constructive method to check, whether these sequences have a monotone (m + 1)-block factor representation

ON REGRESSION REPRESENTATIONS OF STOCHASTIC-PROCESSES

AbstractWe construct a.s. nonlinear regression representations of general stochastic processes (Xn)nϵN. As a consequence we obtain in particular special regression representations of Markov chains and of certain m-dependent sequences. For m-dependent sequences we obtain a constructive method to check, whether these sequences have a monotone (m+1)-block factor representation

Random fractals and Probability metrics

New metrics are introduced in the space of random measures and are applied, with various modifications of the contraction method, to prove existence and uniqueness results for self-similar random fractal measures. We obtain exponential convergence, both in distribution and almost surely, of an iterative sequence of random measures (defined by means of the scaling operator) to a unique self-similar random measure. The assumptions are quite weak, and correspond to similar conditions in the deterministic case. The fixed mass case is handled in a direct way based on regularity properties of the metrics and the properties of a natural probability space. Proving convergence in the random mass case needs additional tools, such as a specially adapted choice of the space of random measures and of the space of probability distributions on measures, the introduction of reweighted sequences of random measures and a comparison technique