Intersection local times of independent fractional Brownian motions as generalized white noise functionals

In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and more general Hurst coefficients.Comment: 28 page

Multi-dimensional parameter estimation of heavy-tailed moving averages

In this paper we present a parametric estimation method for certain multi-parameter heavy-tailed L\'evy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained in [3] via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to the papers [14, 15], which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving L\'evy motion. We extend their work to allow for a multi-parametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method

Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws

We define a large class of continuous time multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk (MRW) [Bacry-Delour-Muzy] and the log-Poisson "product of cynlindrical pulses" [Barral-Mandelbrot]. Our construction is based on some ``continuous stochastic multiplication'' from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide some methods for numerical simulations and discuss some specific examples.Comment: 24 pages, 4 figure

Chromosomal contacts connect loci associated with autism, BMI and head circumference phenotypes

Copy number variants (CNVs) are major contributors to genomic imbalance disorders. Phenotyping of 137 unrelated deletion and reciprocal duplication carriers of the distal 16p11.2 220 kb BP2-BP3 interval showed that these rearrangements are associated with autism spectrum disorders and mirror phenotypes of obesity/underweight and macrocephaly/microcephaly. Such phenotypes were previously associated with rearrangements of the non-overlapping proximal 16p11.2 600 kb BP4-BP5 interval. These two CNV-prone regions at 16p11.2 are reciprocally engaged in complex chromatin looping, as successfully confirmed by 4C-seq, fluorescence in situ hybridization and Hi-C, as well as coordinated expression and regulation of encompassed genes. We observed that genes differentially expressed in 16p11.2 BP4-BP5 CNV carriers are concomitantly modified in their chromatin interactions, suggesting that disruption of chromatin interplays could participate in the observed phenotypes. We also identified cis- and trans-acting chromatin contacts to other genomic regions previously associated with analogous phenotypes. For example, we uncovered that individuals with reciprocal rearrangements of the trans-contacted 2p15 locus similarly display mirror phenotypes on head circumference and weight. Our results indicate that chromosomal contacts’ maps could uncover functionally and clinically related genes.Molecular Psychiatry advance online publication, 31 May 2016; doi:10.1038/mp.2016.84

has attracted interest. For a motivation and further references, we refer the reader to Barczy and Pap

Abstract: Let α,T> 0. We study the asymptotic properties of a least squares estimator for the parameter α of a fractional bridge defined as dXt = −α Xt T−t dt + dBt, 0 � t < T, where B is a fractional Brownian motion of Hurst parameter H> 1 2. Depending on the value of α, we prove that we may have strong consistency or not as t → T. When we have consistency, we obtain the rate of this convergence as well. Also, we compare our results to the (known) case where B is replaced by a standard Brownian motion W. It is great pleasure for us to dedicate this paper to our friend David Nualart, in celebration of his 60th birthday and with all our admiration. hal-00560815, version 3- 2 Aug 2013