Recognition of stable distribution with Lévy index α close to 2.

We address the problem of recognizing α-stable Lévy distribution with Lévy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Lévy to Gaussian fluctuation statistics.

[1]  R. Sánchez,et al.  Kinetic equation of linear fractional stable motion and applications to modeling the scaling of intermittent bursts. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  V. S. Romanov,et al.  Spectral and statistical analysis of fluctuations in the SOL and diverted plasmas of the Uragan-3M torsatron , 2009 .

[3]  B. M. Brown,et al.  Normal Scores, Normal Plots Tests for Normality , 1996 .

[4]  P. Barthelemy,et al.  A Lévy flight for light , 2008, Nature.

[5]  S. Chapman,et al.  Characterization and interpretation of strongly nonlinear phenomena in fusion, space and astrophysical plasmas , 2006 .

[6]  J. McCulloch,et al.  Simple consistent estimators of stable distribution parameters , 1986 .

[7]  Ralph B. D'Agostino,et al.  Goodness-of-Fit-Techniques , 2020 .

[8]  K. Burnecki,et al.  Fractional Lévy stable motion can model subdiffusive dynamics. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Yuichi Mori,et al.  Handbook of computational statistics : concepts and methods , 2004 .

[10]  V. Demidov,et al.  Electric probes for plasmas: The link between theory and instrument , 2002 .

[11]  T. Geisel,et al.  The scaling laws of human travel , 2006, Nature.

[12]  W. DuMouchel.  On the Asymptotic Normality of the Maximum-Likelihood Estimate when Sampling from a Stable Distribution , 1973 .

[13]  I. A. Koutrouvelis.  An iterative procedure for the estimation of the parameters of stable laws , 1981 .

[14]  R. Mantegna,et al.  An Introduction to Econophysics: Contents , 1999 .

[15]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[16]  Ioannis A. Koutrouvelis,et al.  Regression-Type Estimation of the Parameters of Stable Laws , 1980 .

[17]  I. Sokolov,et al.  Anomalous transport : foundations and applications , 2008 .

[18]  Jeffrey M. Hausdorff,et al.  Long-range anticorrelations and non-Gaussian behavior of the heartbeat. , 1993, Physical review letters.

[19]  P D Ditlevsen.  Anomalous jumping in a double-well potential. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  R. Weron.  Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime , 2001, cond-mat/0103256.

[21]  Anil K. Bera,et al.  A test for normality of observations and regression residuals , 1987 .

[22]  P. Lee,et al.  14. Simulation and Chaotic Behaviour of α‐Stable Stochastic Processes , 1995 .

[23]  N. N. Skvortsova,et al.  Stochastic Models of Structural Plasma Turbulence , 2006 .

[24]  Edge fluctuation studies in Heliotron J , 2005 .

[25]  A. Weron,et al.  Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes , 1993 .

[26]  Rosario N. Mantegna,et al.  An Introduction to Econophysics: Contents , 1999 .

[27]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[28]  C. L. Nikias,et al.  Signal processing with alpha-stable distributions and applications , 1995 .

[29]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[30]  B. Gnedenko,et al.  Limit Distributions for Sums of Independent Random Variables , 1955 .

[31]  P. Levy,et al.  Calcul des Probabilites , 1926, The Mathematical Gazette.

[32]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[33]  Nicolas E. Humphries,et al.  Scaling laws of marine predator search behaviour , 2008, Nature.

[34]  K. Knight.  Stable Non-Gaussian Random Processes Gennady Samorodnitsky and Murad S. Taqqu Chapman and Hall, 1994 , 1997, Econometric Theory.

[35]  Agnieszka Wyłomańska,et al.  Subdynamics of financial data from fractional Fokker-Planck equation , 2009 .

[36]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[37]  F. Wagner,et al.  A quarter-century of H-mode studies , 2007 .

[38]  B. M. Hill,et al.  A Simple General Approach to Inference About the Tail of a Distribution , 1975 .

[39]  V. Gonchar,et al.  Stable Lévy distributions of the density and potential fluctuations in the edge plasma of the U-3M torsatron , 2003 .

[40]  A. V. Lozin,et al.  Processes arising in the edge and diverted plasmas during ITB formation in the U-3M torsatron , 2006 .

[41]  D. R. Kulkarni,et al.  Evidence of Lévy stable process in tokamak edge turbulence , 2001 .

[42]  W. Härdle,et al.  Statistical Tools for Finance and Insurance , 2003 .