Stein's method on the second Wiener chaos : 2-Wasserstein distance

In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called Stein equations at the heart of Stein's method. We provide several applications, and discuss comparison with recent similar results on the same topic

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of . We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent

Reciprocal regulation of aurora kinase A and ATIP3 in the control of metaphase spindle length

International audienceMaintaining the integrity of the mitotic spindle in metaphase is essential to ensure normal cell division. We show here that depletion of microtubule-associated protein ATIP3 reduces metaphase spindle length. Mass spectrometry analyses identi-ied the microtubule minus-end depolymerizing kinesin Kif2A as an ATIP3 binding protein. We show that ATIP3 controls metaphase spindle length by interacting with Kif2A and its partner Dda3 in an Aurora kinase A-dependent manner. In the absence of ATIP3, Kif2A and Dda3 accumulate at spindle poles, which is consistent with reduced poleward microtubule lux and shortening of the spindle. ATIP3 silencing also limits Aurora A localization to the poles. Transfection of GFP-Aurora A, but not kinase-dead mutant, rescues the phenotype, indicating that ATIP3 maintains Aurora A activity on the poles to control Kif2A targeting and spindle size. Collectively, these data emphasize the pivotal role of Aurora kinase A and its mutual regulation with ATIP3 in controlling spindle length

Atmospheric Escape Processes and Planetary Atmospheric Evolution

The habitability of the surface of any planet is determined by a complex evolution of its interior, surface, and atmosphere. The electromagnetic and particle radiation of stars drive thermal, chemical and physical alteration of planetary atmospheres, including escape. Many known extrasolar planets experience vastly different stellar environments than those in our Solar system: it is crucial to understand the broad range of processes that lead to atmospheric escape and evolution under a wide range of conditions if we are to assess the habitability of worlds around other stars. One problem encountered between the planetary and the astrophysics communities is a lack of common language for describing escape processes. Each community has customary approximations that may be questioned by the other, such as the hypothesis of H-dominated thermosphere for astrophysicists, or the Sun-like nature of the stars for planetary scientists. Since exoplanets are becoming one of the main targets for the detection of life, a common set of definitions and hypotheses are required. We review the different escape mechanisms proposed for the evolution of planetary and exoplanetary atmospheres. We propose a common definition for the different escape mechanisms, and we show the important parameters to take into account when evaluating the escape at a planet in time. We show that the paradigm of the magnetic field as an atmospheric shield should be changed and that recent work on the history of Xenon in Earth's atmosphere gives an elegant explanation to its enrichment in heavier isotopes: the so-called Xenon paradox

Distances between probability distributions via characteristic functions and biasing

28 pagesIn a spirit close to classical Stein's method, we introduce a new technique to derive first order ODEs on differences of characteristic functions. Then, using concentration inequalities and Fourier transform tools, we convert this information into sharp bounds for the so-called smooth Wasserstein metrics which frequently arise in Stein's method theory. Our methodolgy is particularly efficient when the target density and the object of interest both satisfy biasing equations (including the zero-bias and size-bias mechanisms). In order to illustrate our technique we provide estimates for: (i) the Dickman approximation of the sum of positions of records in random permutations, (ii) quantitative limit theorems for convergence towards stable distributions, (iii) a general class of infinitely divisible distributions and (iv) distributions belonging to the second Wiener chaos, that is to say, in some situations impervious to the standard procedures of Stein's method. Except for the stable distribution, our bounds are sharp up to logarithmic factors. We also describe a general argument and open the way for a wealth of refinements and other applications

A bound on the 2-Wasserstein distance between linear combinations of independent random variables

This preprint corresponds to the third section of https://arxiv.org/abs/1601.03301. The main result and its proof have been completely re-written. We provide new applicationsWe provide a bound on a natural distance between finitely and infinitely supported elements of the unit sphere of $\ell^2(\mathbb{N}^*)$, the space of real valued sequences with finite $\ell^2$ norm. We use this bound to estimate the 2-Wasserstein distance between random variables which can be represented as linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure which is related to Nourdin and Peccati's Malliavin-Stein method. The main area of application of our results is towards the computation of quantitative rates of convergence towards elements of the second Wiener chaos. After particularizing our bounds to this setting and comparing them with the available literature on the subject (particularly the Malliavin-Stein method for Variance-gamma random variables), we illustrate their versatility by tackling three examples: chi-squared approximation for second order $U$-statistics, asymptotics for sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent

Stein characterizations for linear combinations of gamma random variables

This corresponds to the second section of https://arxiv.org/abs/1601.03301 New results added and applications provided. arXiv admin note: text overlap with arXiv:1601.03301In this paper we propose a new, simple and explicit mechanism allowing to derive Stein operators for random variables whose characteristic function satisfies a simple ODE. We apply this to study random variables which can be represented as linear combinations of (non necessarily independent) gamma distributed random variables. The connection with Malliavin calculus for random variables in the second Wiener chaos is detailed. An application to McKay Type I random variables is also outlined

Stein characterizations for linear combinations of gamma random variables

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