On the properties of random multiplicative measures with the multipliers exponentially distributed

Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with their multipliers satisfying exponentially distributed is investigated in detail. Branching emerges in the curve of generalized dimensions, and negative values of generalized dimensions arise. Three equivalent methods of classification of the random multifractal measures are proposed, which is based on: (i) the discrepancy between the curves of generalized dimensions, (ii) the solution properties of equation T(qcrit) =0, and (iii) the relative position of the curve f(alpha) and the diagonal f(alpha)=alpha in the first quadrant. These three classes correspond to \mu([0,1])=infinity, \mu([0,1])=1 and \mu([0,1])=0, respectively. Phase diagram is introduced to illustrate the diverse performance of the random measures that is multiplicatively generated.Comment: 12 pages,7 figures,Revesion of TW060600 submitted to Physica

Estimating long range dependence: finite sample properties and confidence intervals

A major issue in financial economics is the behavior of asset returns over long horizons. Various estimators of long range dependence have been proposed. Even though some have known asymptotic properties, it is important to test their accuracy by using simulated series of different lengths. We test R/S analysis, Detrended Fluctuation Analysis and periodogram regression methods on samples drawn from Gaussian white noise. The DFA statistics turns out to be the unanimous winner. Unfortunately, no asymptotic distribution theory has been derived for this statistics so far. We were able, however, to construct empirical (i.e. approximate) confidence intervals for all three methods. The obtained values differ largely from heuristic values proposed by some authors for the R/S statistics and are very close to asymptotic values for the periodogram regression method.Comment: 16 pages, 11 figures New version: 14 pages (smaller fonts), 11 figures, new Section on application

Fractal geometry of normal phase clusters and magnetic flux trapping in high-Tc superconductors

The effect of geometry and morphology of superconducting structure on magnetic flux trapping is considered. It is found that the clusters of normal phase, which act as pinning centers, have significant fractal properties. The fractal dimension of the boundary of these clusters is estimated using a simple area-perimeter relation. A superconductor is treated as a percolation system. It is revealed that the fractality intensifies the magnetic flux trapping and thereby enhances the critical current value.Comment: 5 pages with 1 table and 2 figures, revtex, published in Phys.Lett.A 267 (2000) 66 with more complicated figure

Generic Multifractality in Exponentials of Long Memory Processes

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent $\phi+1/2$, where $\phi >0$. This generalizes previous studies performed only with $\phi=0$ (with a truncation at an integral scale), by showing that multifractality holds over a remarkably large range of dimensionless scales for $\phi>0$. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation $\phi$ from 1/2 and of another parameter $\sigma^2$ embodying information on the short-range amplitude of the memory kernel, the ultra-violet cut-off (``viscous'') scale and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the ``inertial'' scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra $\zeta(q)$ on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum $\zeta(q)$ by different combinations of $\phi$ and $\sigma^2$.Comment: 10 pages + 9 figure

Latt\`es maps and combinatorial expansion

A Latt\`es map $f\colon \hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism. We characterize Latt\`es maps by their combinatorial expansion behavior.Comment: 41 pages, 3 figures. arXiv admin note: text overlap with arXiv:1109.2980; and with arXiv:1009.3647 by other author

Long-term temporal dependence of droplets transiting through a fixed spatial point in gas-liquid twophase turbulent jets

We perform rescaled range analysis upon the signals measured by Dual Particle Dynamical Analyzer in gas-liquid two-phase turbulent jets. A novel rescaled range analysis is proposed to investigate these unevenly sampled signals. The Hurst exponents of velocity and other passive scalars in the bulk of spray are obtained to be 0.59$\pm$0.02 and the fractal dimension is hence 1.41$\pm$ 0.02, which are in remarkable agreement with and much more precise than previous results. These scaling exponents are found to be independent of the configuration and dimensions of the nozzle and the fluid flows. Therefore, such type of systems form a universality class with invariant scaling properties.Comment: 16 Elsart pages including 8 eps figure

Decomposing Intraday Dependence in Currency Markets: Evidence from the AUD/USD Spot Market

The local Hurst exponent, a measure employed to detect the presence of dependence in a time series, may also be used to investigate the source of intraday variation observed in the returns in foreign exchange markets. Given that changes in the local Hurst exponent may be due to either a time-varying range, or standard deviation, or both of these simultaneously, values for the range, standard deviation and local Hurst exponent are recorded and analyzed separately. To illustrate this approach, a high-frequency data set of the spot Australian dollar/U.S. dollar provides evidence of the returns distribution across the 24-hour trading day with time-varying dependence and volatility clearly aligning with the opening and closing of markets. This variation is attributed to the effects of liquidity and the price-discovery actions of dealers.Comment: 3 Figures, 3 Tables, 28 page

Numerical investigations of discrete scale invariance in fractals and multifractal measures

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, {\em{i.e.}}, no preferred scaling ratio since the set of complex exponents spread irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly.Comment: 31 Elsart pages including 12 eps figure