Gaussian Tails Research Articles

Analysis of Fokker–Planck approach for foreign exchange market statistics study

In a well-known work (Phys. Rev. Lett. 84 (2000) 5224) it was shown that behaviour of returns for foreign exchange markets in different time scales can be described in terms of Fokker–Planck equation, with Kramers–Moyal coefficients being estimated from the empirical data. In the current paper the authors provide analytical solution for stationary Fokker–Planck equation, which allows explanation of non Gaussian tails of the distribution function. It is also shown that while approximating empirical data one needs to observe some limitations for correct results obtaining.

Strong solutions of stochastic equations with singular time dependent drift

We prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L q _L p -integrability of b in ℝ×G with d/p+2/q<1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function ψ blowing up as G∋x→∂G, we prove that the conditions 2D t ψ≤Kψ,2D t ψ+Δψ≤Ke ɛψ ,ɛ ∈ [0,2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.

Development of a Deep Silicon Phase Fresnel Lens Using Gray-Scale Lithography and Deep Reactive Ion Etching

We report the first fabrication and development of a deep phase Fresnel lens (PFL) in silicon through the use of gray-scale lithography and deep-reactive ion etching (DRIE). A Gaussian tail approximation is introduced as a method of predicting the height of photoresist gray levels given the relative amount of transmitted light through a gray-scale optical mask. Device mask design is accomplished through command-line scripting in a CAD tool to precisely define the millions of pixels required to generate the appropriate profile in photoresist. Etch selectivity during DRIE pattern transfer is accurately controlled to produce the desired scaling factor between the photoresist and silicon profiles. As a demonstration of this technology, a 1.6-mm diameter PFL is etched 43 /spl mu/m into silicon with each grating profile designed to focus 8.4 keV photons a distance of 118 m.

Exponential velocity tails in a driven inelastic Maxwell model

The problem of the steady-state velocity distribution in a driven inelastic Maxwell model of shaken granular material is revisited. Numerical solution of the master equation and analytical arguments show that the model has bilateral exponential velocity tails [P(v) approximately e(-|v|/sqrt[D])], where D is the amplitude of the noise. Previous study of this model predicted Gaussian tails [P(v) approximately e(-av(2))].

Enhanced interface repulsion from quenched hard–wall randomness

We consider the harmonic crystal, or massless free field, , , that is the centered Gaussian field with covariance given by the Green function of the simple random walk on ℤ d . Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition to be larger than , is an IID field (which is also independent of ϕ), for every x in a large region , with N a positive integer and D a bounded subset of ℝ d . We are mostly motivated by results for given typical realizations of σ (quenched set–up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d+1)–dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ 0 is heavier than Gaussian, while essentially no effect is observed if the tail is sub–Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ, leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as of the probability that ϕ lies above σ in D N .

Gaussian tail for empirical distributions of MST on random graphs

Consider the complete graph K n on n vertices and the n-cube graph Q n on 2 n vertices. Suppose independent uniform random edge weights are assigned to each edges in K n and Q n and let T(K n) and T(Q n) denote the unique minimal spanning trees on K n and Q n , respectively. In this paper we obtain the Gaussian tail for the number of edges of T(K n) and T(Q n) with weight at most t/ n.

Large deviations and mean-field theory for asymmetric random recurrent neural networks

In this article, we study the asymptotic dynamics of a noisy discrete time neural network, with random asymmetric couplings and thresholds. More precisely, we focus our interest on the limit behaviour of the network when its size grows to infinity with bounded time. In the case of gaussian connection weights, we use the same techniques as Ben Arous and Guionnet (see [3]) to prove that the image law of the distribution of the neurons' activation states by the empirical measure satisfies a temperature free large deviation principle. Moreover, we prove that if the connection weights satisfy a general condition of domination by gaussian tails, then the distribution of the activation potential of each neuron converges weakly towards an explicit gaussian law, the characteristics of which are contained in the mean-field equations stated by Cessac-Doyon-Quoy-Samuelides (see [4–6]). Furthermore, under this hypothesis, we obtain a law of large numbers and a propagation of chaos result. Finally, we show that many classical distributions on the couplings fulfill our general condition. Thus, this paper provides rigorous mean-field results for a large class of neural networks which is currently investigated in neural network literature.

Driven inelastic Maxwell models with high energy tails.

The solutions of the homogeneous nonlinear Boltzmann equation for inelastic Maxwell models, when driven by different types of thermostats, show, in general, overpopulated high energy tails of the form approximately exp(-ac), with power law tails and Gaussian tails as border line cases. The results are compared with those for inelastic hard spheres, and a comprehensive picture of the long time behavior in freely cooling and driven inelastic systems is presented.

On asymptotics of multivariate integrals with applications to records

Let { X n ,n≥1} be a sequence of iid. Gaussian random vectors in R d , d≥2, with nonsingular distribution function F. In this paper the asymptotics for the sequence of integrals I F,n (G n )≔n∫ R d G n n−1( X ) dF( X ) is considered with G n some distribution function on R d . In the case G n =F the integral I F,n (F)/n is the probability that a record occurs in X 1,…, X n at index n. [1] obtained lower and upper asymptotic bounds for this case, whereas [2] showed the rate of convergence if d=2. In this paper we derive the exact rate of convergence of I F,n (G n ) for d≥2 under some restrictions on the distribution function G n . Some related results for multivariate Gaussian tails are discussed also.

Nontrivial velocity distributions in inelastic gases

We study freely evolving and forced inelastic gases using the Boltzmann equation. We consider uniform collision rates and obtain analytical results valid for arbitrary spatial dimension d and arbitrary dissipation coefficient epsilon. In the freely evolving case, we find that the velocity distribution decays algebraically, P(v,t) ~ v^{-sigma} for sufficiently large velocities. We derive the exponent sigma(d,epsilon), which exhibits nontrivial dependence on both d and epsilon, exactly. In the forced case, the velocity distribution approaches a steady-state with a Gaussian large velocity tail.

Analytical solution of a model for complex food webs.

We investigate numerically and analytically a recently proposed model for food webs [Nature 404, 180 (2000)] in the limit of large web sizes and sparse interaction matrices. We obtain analytical expressions for several quantities with ecological interest, in particular, the probability distributions for the number of prey and the number of predators. We find that these distributions have fast-decaying exponential and Gaussian tails, respectively. We also find that our analytical expressions are robust to changes in the details of the model.

Optical Properties of PZT 65/35 Thin Films Deposited by Sol-Gel

PZT 65/35 thin films were deposited on Pt-coated Si (100) and MgO (110) substrates using the sol-gel technique. The crystalline structure of the films was investigated by X-ray diffraction and Raman spectroscopy. Results of measurements and modeling of the optical spectra for the PZT films on MgO substrates are reported. The effective dielectric function is calculated versus frequency and Gaussian tail of the join density of states under the assumption of direct band to band transition.

Form Factors of Block Copolymer Micelles with Excluded-Volume Interactions of the Corona Chains Determined by Monte Carlo Simulations

The scattering of a diblock copolymer micelle has been simulated using Monte Carlo techniques. The scattering is analyzed using a novel model, where the corona is represented as a dilute/semidilute polymer solution with a radial profile. This approach decouples the scattering due to interaction and connectivity induced density fluctuations and scattering due to the radial profile of the corona. Three different profiles have been used to fit the simulated corona scattering: a box with a Gaussian tail and two maximum entropy (ME) profiles; chain penetration into the core region is not allowed for any of the profiles. Excellent fits are obtained, especially for a ME profile with three parameters. An excluded-volume parameter and the corona compressibility are obtained and show a strong dependence on surface coverage. The derived expressions for the form factor provides a new approach for analyzing experimental data obtained by neutron or X-ray small-angle scattering for block copolymer micelles with significant intra- and interchain excluded-volume interactions.

Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case

We establish the exact low-energy asymptotics of the integrated density of states (Lifschitz tail) in a homogeneous magnetic field and Poissonian impurities with a repulsive single-site potential of Gaussian decay. It has been known that the Gaussian potential tail discriminates between the so-called “classical” and “quantum” regimes, and precise asymptotics are known in these cases. For the borderline case, the coexistence of the classical and quantum regimes was conjectured. Here we settle this last remaining open case to complete the full picture of the magnetic Lifschitz tails.

Non‐Gaussian invariant measures for the Majda model of decaying turbulent transport

AbstractThe problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature are fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large deviation problem for the scalar field into a small deviation, or small ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here, such a methodology is applied to a model proposed by Majda consisting of a white‐in‐time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non‐Gaussian invariant measure for the (reduced) scalar field is derived and, in particular, it is shown that the one‐point distribution of the scalar has stretched exponential tails, with a stretching exponent depending of the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one‐to‐one correspondence with a exponent measuring time persistence effects in the velocity field. © 2001 John Wiley &amp; Sons, Inc.

An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random Variables

We provide precise bounds for tail probabilities, say ℙ{M n ⩾ x}, of sums M n of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed.

Optical Solitons with Gaussian Tails**The project supported by National Natural Science Foundation of China, the Development Foundation for Science and Technology of ECNU, the Hong Kong Research Grant Council and the Hong Kong Baptist University Faculty Research Grant

Based on an extended method of multiple scales, an analytical description of the optical solitons in a model for the propagation of nonlinear pulses in dispersion-managed optical fibers is presented. The results show that the optical pulses can be expressed by quantummechanical wavefunctions of a linear oscillator, which have tails of Gaussian type, modulated by hyperbolic secant solitons of the nonlinear Schrödinger equation.

Lattice models of advection-diffusion.

We present a synthesis of theoretical results concerning the probability distribution of the concentration of a passive tracer subject to both diffusion and to advection by a spatially smooth time-dependent flow. The freely decaying case is contrasted with the equilibrium case. A computationally efficient model of advection-diffusion on a lattice is introduced, and used to test and probe the limits of the theoretical ideas. It is shown that the probability distribution for the freely decaying case has fat tails, which have slower than exponential decay. The additively forced case has a Gaussian core and exponential tails, in full conformance with prior theoretical expectations. An analysis of the magnitude and implications of temporal fluctuations of the conditional diffusion and dissipation is presented, showing the importance of these fluctuations in governing the shape of the tails. Some results concerning the probability distribution of dissipation, and concerning the spatial scaling properties of concentration fluctuation, are also presented. Though the lattice model is applied only to smooth flow in the present work, it is readily applicable to problems involving rough flow, and to chemically reacting tracers. (c) 2000 American Institute of Physics.

The Multivariate Gaussian Tail Model: An Application to Oceanographic Data

SUMMARY Optimal design of sea-walls requires the extreme value analysis of a variety of oceanographic data. Asymptotic arguments suggest the use of multivariate extreme value models, but empirical studies based on data from several UK locations have revealed an inadequacy of this class for modelling the types of dependence that are often encountered in such data. This paper develops a specific model based on the marginal transformation of the tail of a multivariate Gaussian distribution and examines its utility in overcoming the limitations that are encountered with the current methodology. Diagnostics for the model are developed and the robustness of the model is demonstrated through a simulation study. Our analysis focuses on extreme sea-levels at Newlyn, a port in south-west England, for which previous studies had given conflicting estimates of the probability of flooding. The novel diagnostics suggest that this discrepancy may be due to the weak dependence at extreme levels between wave periods and both wave heights and still water levels. The multivariate Gaussian tail model is shown to resolve the conflict and to offer a convincing description of the extremal sea-state process at Newlyn.

Rigorous Estimates of the Tails of the Probability Distribution Function for the Random Linear Shear Model

In previous work Majda and McLaughlin, and Majda computed explicit expressions for the 2Nth moments of a passive scalar advected by a linear shear flow in the form of an integral over R N . In this paper we first compute the asymptotics of these moments for large moment number. We are able to use this information about the large-N behavior of the moments, along with some basic facts about entire functions of finite order, to compute the asymptotics of the tails of the probability distribution function. We find that the probability distribution has Gaussian tails when the energy is concentrated in the largest scales. As the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer, and the distribution moves smoothly from Gaussian through exponential and “stretched exponential.” We also show that the derivatives of the scalar are increasingly intermittent, in agreement with experimental observations, and relate the exponents of the scalar derivative to the exponents of the scalar.