4,613 research outputs found
Non-Gaussian Distributions in Extended Dynamical Systems
We propose a novel mechanism for the origin of non-Gaussian tails in the
probability distribution functions (PDFs) of local variables in nonlinear,
diffusive, dynamical systems including passive scalars advected by chaotic
velocity fields. Intermittent fluctuations on appropriate time scales in the
amplitude of the (chaotic) noise can lead to exponential tails. We provide
numerical evidence for such behavior in deterministic, discrete-time passive
scalar models. Different possibilities for PDFs are also outlined.Comment: 12 pages and 6 figs obtainable from the authors, LaTex file,
OSU-preprint-
Microscopic Origin of Non-Gaussian Distributions of Financial Returns
In this paper we study the possible microscopic origin of heavy-tailed
probability density distributions for the price variation of financial
instruments. We extend the standard log-normal process to include another
random component in the so-called stochastic volatility models. We study these
models under an assumption, akin to the Born-Oppenheimer approximation, in
which the volatility has already relaxed to its equilibrium distribution and
acts as a background to the evolution of the price process. In this
approximation, we show that all models of stochastic volatility should exhibit
a scaling relation in the time lag of zero-drift modified log-returns. We
verify that the Dow-Jones Industrial Average index indeed follows this scaling.
We then focus on two popular stochastic volatility models, the Heston and
Hull-White models. In particular, we show that in the Hull-White model the
resulting probability distribution of log-returns in this approximation
corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are
given in terms of the microscopic stochastic volatility model. Finally, we show
that the log-returns for 30 years Dow Jones index data is well fitted by a
Tsallis distribution, obtaining the relevant parameters.Comment: 13 pages, 4 figures. Several clarifying comments, new references and
acknowledgments adde
Non-Gaussian distributions under scrutiny
International audienceComment of the very interesting paper by Hilhorst & Schehr, J. Stat. Mech. P06003 (2007). The main point is that one should be extremely careful when interpreting non-Gaussian data in terms of q-Gaussians
ECA: High Dimensional Elliptical Component Analysis in non-Gaussian Distributions
We present a robust alternative to principal component analysis (PCA) ---
called elliptical component analysis (ECA) --- for analyzing high dimensional,
elliptically distributed data. ECA estimates the eigenspace of the covariance
matrix of the elliptical data. To cope with heavy-tailed elliptical
distributions, a multivariate rank statistic is exploited. At the model-level,
we consider two settings: either that the leading eigenvectors of the
covariance matrix are non-sparse or that they are sparse. Methodologically, we
propose ECA procedures for both non-sparse and sparse settings. Theoretically,
we provide both non-asymptotic and asymptotic analyses quantifying the
theoretical performances of ECA. In the non-sparse setting, we show that ECA's
performance is highly related to the effective rank of the covariance matrix.
In the sparse setting, the results are twofold: (i) We show that the sparse ECA
estimator based on a combinatoric program attains the optimal rate of
convergence; (ii) Based on some recent developments in estimating sparse
leading eigenvectors, we show that a computationally efficient sparse ECA
estimator attains the optimal rate of convergence under a suboptimal scaling.Comment: to appear in JASA (T&M
Stochastic Calculus for Assets with Non-Gaussian Price Fluctuations
From the path integral formalism for price fluctuations with non-Gaussian
distributions I derive the appropriate stochastic calculus replacing Ito's
calculus for stochastic fluctuations.Comment: Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of
paper (including all PS fonts) at
http://www.physik.fu-berlin.de/~kleinert/32
Statistical properties of solar wind discontinuities, intermittent turbulence, and rapid emergence of non-Gaussian distributions
Recent studies have compared properties of the magnetic field in simulations of Hall MHD turbulence with spacecraft data, focusing on methods used to identify classical discontinuities and intermittency statistics. Comparison of ACE solar wind data and simulations of 2D and 3D turbulence shows good agreement in waiting‐time analysis of magnetic discontinuities, and in the related distribution of magnetic field increments. This supports the idea that the magnetic structures in the solar wind may emerge fast and locally from nonlinear dynamics that can be understood in the framework of nonlinear MHD theory. The analysis suggests that small scale current sheets form spontaneously and rapidly enough that some of the observed solar wind discontinuities may be locally generated, representing boundaries between interacting flux tubes
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