Estimator of a non-Gaussian parameter in multiplicative log-normal models.

We study non-Gaussian probability density functions (PDF's) of multiplicative log-normal models in which the multiplication of Gaussian and log-normally distributed random variables is considered. To describe the PDF of the velocity difference between two points in fully developed turbulent flows, the non-Gaussian PDF model was originally introduced by Castaing [Physica D 46, 177 (1990)]. In practical applications, an experimental PDF is approximated with Castaing's model by tuning a single non-Gaussian parameter, which corresponds to the logarithmic variance of the log-normally distributed variable in the model. In this paper, we propose an estimator of the non-Gaussian parameter based on the q th order absolute moments. To test the estimator, we introduce two types of stochastic processes within the framework of the multiplicative log-normal model. One is a sequence of independent and identically distributed random variables. The other is a log-normal cascade-type multiplicative process. By analyzing the numerically generated time series, we demonstrate that the estimator can reliably determine the theoretical value of the non-Gaussian parameter. Scale dependence of the non-Gaussian parameter in multiplicative log-normal models is also studied, both analytically and numerically. As an application of the estimator, we demonstrate that non-Gaussian PDF's observed in the S&P500 index fluctuations are well described by the multiplicative log-normal model.

[1]  Y. Gagne,et al.  Velocity probability density functions of high Reynolds number turbulence , 1990 .

[2]  Emmanuel Bacry,et al.  Random cascades on wavelet dyadic trees , 1998 .

[3]  A. M. Oboukhov.  Some specific features of atmospheric tubulence , 1962, Journal of Fluid Mechanics.

[4]  B Rubinsky,et al.  On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity , 1992 .

[5]  E. Bacry,et al.  Modelling fluctuations of financial time series: from cascade process to stochastic volatility model , 2000, cond-mat/0005400.

[6]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[7]  M. Ausloos,et al.  Low q-moment multifractal analysis of Gold price, Dow Jones Industrial Average and BGL-USD exchange rate , 1999 .

[8]  Zbigniew R Struzik,et al.  Phase transition in a healthy human heart rate. , 2005, Physical review letters.

[9]  R. Hughson,et al.  Modeling heart rate variability in healthy humans: a turbulence analogy. , 2001, Physical review letters.

[10]  Nobuko Fuchikami,et al.  Dynamical model of financial markets: fluctuating 'temperature' causes intermittent behavior of price changes , 2003 .

[11]  Junichiro Hayano,et al.  Critical scale invariance in a healthy human heart rate. , 2004, Physical review letters.

[12]  H. Nakao.  Multi-scaling properties of truncated Lévy flights , 2000, cond-mat/0002027.

[13]  Zbigniew R Struzik,et al.  Criticality and phase transition in stock-price fluctuations. , 2006, Physical review letters.

[14]  E. Bacry,et al.  Multifractal random walk. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  J. Peinke,et al.  Turbulent cascades in foreign exchange markets , 1996, Nature.

[16]  David W. Scott,et al.  Multivariate Density Estimation: Theory, Practice, and Visualization , 1992, Wiley Series in Probability and Statistics.

[17]  Peinke,et al.  Transition toward developed turbulence. , 1994, Physical review letters.

[18]  M Ausloos,et al.  Dynamical model and nonextensive statistical mechanics of a market index on large time windows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Grossmann,et al.  Effect of dissipation fluctuations on anomalous velocity scaling in turbulence. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[20]  D. Sornette,et al.  ”Direct” causal cascade in the stock market , 1998 .

[21]  Vincenzo Carbone,et al.  Intermittency in the solar wind turbulence through probability distribution functions of fluctuations , 1999, physics/9903043.

[22]  Roberto Benzi,et al.  A random process for the construction of multiaffine fields , 1993 .

[23]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[24]  A. Kolmogorov.  A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number , 1962, Journal of Fluid Mechanics.

[25]  Robert F. Cahalan,et al.  Bounded cascade models as nonstationary multifractals. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.