scholarly journals A simple estimator for the characteristic exponent of the stable Paretian distribution

1999 ◽  
Vol 29 (10-12) ◽  
pp. 161-176 ◽  
Author(s):  
S. Mittnik ◽  
M.S. Paolella
Keyword(s):  
Characteristic Exponent ◽  
Stable Paretian Distribution

COLONY OF THE FUNGUS Aspergillus oryzae AND SELF-AFFINE FRACTAL GEOMETRY OF GROWTH FRONTS

Fractals ◽  
1993 ◽  
Vol 01 (01) ◽  
pp. 11-19 ◽  
Author(s):  
SHU MATSUURA ◽  
SASUKE MIYAZIMA
Keyword(s):  
Aspergillus Oryzae ◽  
Environmental Conditions ◽  
Fractal Geometry ◽  
Nutrient Concentration ◽  
Incubation Temperature ◽  
Characteristic Exponent ◽  
Growth Front ◽  
Favorable Condition ◽  
The Self ◽  
The Mean

A variety of colony shapes of the fungus Aspergillus oryzae under varying environmental conditions such as the nutrient concentration, medium stiffness and incubation temperature are obtained, ranging from a homogeneous Eden-like to a ramified DLA-like pattern. The roughness σ(l, h) of the growth front of the band-shaped colony, where h is the mean front height within l of the horizontal range, satisfies the self-affine fractal relation under favorable environmental conditions. In the most favorable condition of our experiments, its characteristic exponent is found to be a little larger than that of the 2-dimensional Eden model.

scholarly journals The Alpha stable distribution in ocean ambient noise modelling

2019 ◽  
Vol 283 ◽  
pp. 08002
Author(s):  
Guoli Song ◽  
Xinyi Guo ◽  
Li Ma
Keyword(s):  
Normal Distribution ◽  
Ambient Noise ◽  
Stable Distribution ◽  
Characteristic Exponent ◽  
Statistical Modelling ◽  
Gaussian State ◽  
Sample Mean ◽  
Noise Interference ◽  
Interference Noise ◽  
Non Gaussian

In view of the non-Gaussian of ocean ambient noise, the  stable distribution is applied to the statistical modelling. Firstly, the one-to-one correspondence between the four parameters of stable distribution and the sample mean, variance, skewness and kurtosis are established according to physical meaning. Then, numerical simulations are conducted to analyze the suitability of stable distribution for non-Gaussian ambient noise. In the case of white noise interference, noise is divided into Gaussian state, leptokurtic, and platykurtic separately. The parameters of stable distribution are estimated by the sample quantile and characteristic function method jointly. The simulation results show that, in the Gaussian state,  stable distribution is equivalent to normal distribution. As for leptokurtic distribution, stable distribution is much better than normal distribution, indicating absolute predominance in impulse-like data modeling. But it is not adaptive for low kurtosis state because its characteristic exponent can’t be bigger than two. Finally, the result is verified by ambient noise collected in three environmental conditions, such as quiet ambient noise, airgun interference noise and ship noise. In all three cases,  stable distribution shows good adaptability and accuracy, especially for the airgun dataset it is far superior to normal distribution.

A Note on Normal Attraction to a Stable Law

1971 ◽  
Vol 14 (3) ◽  
pp. 451-452
Author(s):  
M. V. Menon ◽  
V. Seshadri
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Stable Law ◽  
Common Distribution ◽  
Normal Attraction ◽  
Common Distribution Function ◽  
The Common ◽  
Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

Local dimension for piecewise monotonic maps on the interval

1995 ◽  
Vol 15 (6) ◽  
pp. 1119-1142 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Positive Characteristic ◽  
Characteristic Exponent ◽  
Local Dimension ◽  
Pressure Function ◽  
Piecewise Monotonic ◽  
Almost Everywhere ◽  
Conformal Measures

AbstractThe local dimension of invariant and conformal measures for piecewise monotonic transformations on the interval is considered. For ergodic invariant measures m with positive characteristic exponent χm we show that the local dimension exists almost everywhere and equals hm/χm For certain conformal measures we show a relation between a pressure function and the Hausdorff dimension of sets, on which the local dimension is constant.

scholarly journals Optimal heavy tail estimation – Part 1: Order selection

2017 ◽  
Vol 24 (4) ◽  
pp. 737-744 ◽  
Author(s):  
Manfred Mudelsee ◽  
Miguel A. Bermejo
Keyword(s):  
Time Series ◽  
Heavy Tails ◽  
Characteristic Exponent ◽  
Tail Probability ◽  
Heavy Tail ◽  
Infinite Variance ◽  
Accurate Estimation ◽  
Monte Carlo Experiment ◽  
River Elbe ◽  
Tail Estimation

Abstract. The tail probability, P, of the distribution of a variable is important for risk analysis of extremes. Many variables in complex geophysical systems show heavy tails, where P decreases with the value, x, of a variable as a power law with a characteristic exponent, α. Accurate estimation of α on the basis of data is currently hindered by the problem of the selection of the order, that is, the number of largest x values to utilize for the estimation. This paper presents a new, widely applicable, data-adaptive order selector, which is based on computer simulations and brute force search. It is the first in a set of papers on optimal heavy tail estimation. The new selector outperforms competitors in a Monte Carlo experiment, where simulated data are generated from stable distributions and AR(1) serial dependence. We calculate error bars for the estimated α by means of simulations. We illustrate the method on an artificial time series. We apply it to an observed, hydrological time series from the River Elbe and find an estimated characteristic exponent of 1.48 ± 0.13. This result indicates finite mean but infinite variance of the statistical distribution of river runoff.

scholarly journals The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

2019 ◽  
Vol 169 (1) ◽  
pp. 59-77
Author(s):  
Loïc Chaumont ◽  
Jacek Małecki
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Time Variable ◽  
Stable Processes ◽  
The Laplace Transform ◽  
Entrance Law ◽  
Generalized Eigenfunctions ◽  
Positive Half ◽  
Integral Formulae

Abstract We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

scholarly journals Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

2010 ◽  
Vol 47 (04) ◽  
pp. 1023-1033 ◽  
Author(s):  
A. Kuznetsov
Keyword(s):  
Lévy Processes ◽  
Series Representation ◽  
Theta Functions ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transcendental Equation ◽  
Simple Expression ◽  
Lévy Measure ◽  
Infinite Products ◽  
Wide Range

In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

Non-Uniform Chaotic Dynamics with Implications to Information Processing

1983 ◽  
Vol 38 (11) ◽  
pp. 1157-1169 ◽  
Author(s):  
J. S. Nicolis ◽  
G. Meyer-Kress ◽  
G. Haubs
Keyword(s):  
Information Processing ◽  
Characteristic Exponent ◽  
Quantitative Measure ◽  
External Noise ◽  
Strange Attractors ◽  
Pattern Perception ◽  
Small Subset ◽  
Chaotic Flows ◽  
Lyapunov Characteristic Exponent ◽  
High Degree

We study a new parameter - the "Non-Uniformity Factor" (NUF) -, which we have introduced in [1]. by way of estimating and comparing the deviation from average behavior (expressed by such factors as the Lyapunov characteristic exponent(s) and the information dimension) in various strange attractors (discrete and chaotic flows). Our results show for certain values of the control parameters the inadequacy of the above averaging properties in representing what is actually going on - especially when the strange attractors are employed as dynamical models for information processing and pattern recognition. In such applications (like for example visual pattern perception or communication via a burst-error channel) the high degree of adherence of the processor to a rather small subset of crucial features of the pattern under investigation or the flow, has been documented experimentally: Hence the weakness of concepts such as the entropy in giving in such cases a quantitative measure of the information transaction between the pattern and the processor. We finally investigate the influence of external noise in modifying the NUF

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