The Weierstrass-Mandelbrot Process Provides a Series Approximation to the Harmonizable Fractional Stable Motion

2000 ◽  
pp. 161-179
Author(s):  
Vladas Pipiras ◽  
Murad S. Taqqu
Keyword(s):  
Stable Motion ◽  
Series Approximation ◽  
Fractional Stable Motion

A novel crowd flow model based on linear fractional stable motion

2016 ◽  
Vol 30 (09) ◽  
pp. 1650049 ◽  
Author(s):  
Juan Wei ◽  
Hong Zhang ◽  
Zhenya Wu ◽  
Junlin He ◽  
Yangyong Guo
Keyword(s):  
Flow Rate ◽  
Flow Model ◽  
Threshold Value ◽  
Evacuation Time ◽  
Indoor Space ◽  
Stable Motion ◽  
Crowd Density ◽  
Linear Fractional Stable Motion ◽  
Positive Correlations ◽  
Fractional Stable Motion

For the evacuation dynamics in indoor space, a novel crowd flow model is put forward based on Linear Fractional Stable Motion. Based on position attraction and queuing time, the calculation formula of movement probability is defined and the queuing time is depicted according to linear fractal stable movement. At last, an experiment and simulation platform can be used for performance analysis, studying deeply the relation among system evacuation time, crowd density and exit flow rate. It is concluded that the evacuation time and the exit flow rate have positive correlations with the crowd density, and when the exit width reaches to the threshold value, it will not effectively decrease the evacuation time by further increasing the exit width.

PATH PROPERTIES OF THE LINEAR MULTIFRACTIONAL STABLE MOTION

Fractals ◽  
2005 ◽  
Vol 13 (02) ◽  
pp. 157-178 ◽  
Author(s):  
STILIAN STOEV ◽  
MURAD S. TAQQU
Keyword(s):  
Heavy Tails ◽  
Hölder Regularity ◽  
Self Similarity ◽  
Similarity Parameter ◽  
Stable Motion ◽  
Multifractional Brownian Motion ◽  
Sample Paths ◽  
Hölder Exponents ◽  
Fractional Stable Motion ◽  
Holder Regularity

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t). We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Hölder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Hölder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Hölder exponents. The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Hölder regularity of their paths.

scholarly journals OCCUPATION TIME FLUCTUATION LIMITS OF INFINITE VARIANCE EQUILIBRIUM BRANCHING SYSTEMS

2009 ◽  
Vol 12 (04) ◽  
pp. 593-612 ◽  
Author(s):  
PIOTR MIŁOŚ
Keyword(s):  
Particle System ◽  
Random Measure ◽  
Domain Of Attraction ◽  
Interesting Case ◽  
Occupation Time ◽  
Infinite Variance ◽  
Dependence Structure ◽  
Branching Particle System ◽  
Stable Motion ◽  
Fractional Stable Motion

We establish limit theorems for the fluctuations of the rescaled occupation time of a (d, α, β)-branching particle system. It consists of particles moving according to a symmetric α-stable motion in ℝd. The branching law is in the domain of attraction of a (1 + β)-stable law and the initial condition is the equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate α/β < d < (1 + β)α/β, critical d = (1 + β)α/β and large d > (1 + β)α/β dimensions. In the most interesting case of intermediate dimensions we obtain a version of a fractional stable motion. The long-range dependence structure of this process is also studied. Contrary to this case, limit processes in critical and large dimensions have independent increments.

SIMULATION METHODS FOR LINEAR FRACTIONAL STABLE MOTION AND FARIMA USING THE FAST FOURIER TRANSFORM

Fractals ◽  
2004 ◽  
Vol 12 (01) ◽  
pp. 95-121 ◽  
Author(s):  
STILIAN STOEV ◽  
MURAD S. TAQQU
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Moving Average ◽  
Approximation Error ◽  
Telecommunication Networks ◽  
Stable Motion ◽  
Riemann Sum ◽  
Linear Fractional Stable Motion ◽  
Fractional Stable Motion

We present efficient methods for simulation, using the Fast Fourier Transform (FFT) algorithm, of two classes of processes with symmetric α-stable (SαS) distributions. Namely, (i) the linear fractional stable motion (LFSM) process and (ii) the fractional autoregressive moving average (FARIMA) time series with SαS innovations. These two types of heavy-tailed processes have infinite variances and long-range dependence and they can be used in modeling the traffic of modern computer telecommunication networks. We generate paths of the LFSM process by using Riemann-sum approximations of its SαS stochastic integral representation and paths of the FARIMA time series by truncating their moving average representation. In both the LFSM and FARIMA cases, we compute the involved sums efficiently by using the Fast Fourier Transform algorithm and provide bounds and/or estimates of the approximation error. We discuss different choices of the discretization and truncation parameters involved in our algorithms and illustrate our method. We include MATLAB implementations of these simulation algorithms and indicate how the practitioner can use them.

Stochastic properties of the linear multifractional stable motion

2004 ◽  
Vol 36 (04) ◽  
pp. 1085-1115 ◽  
Author(s):  
Stilian Stoev ◽  
Murad S. Taqqu
Keyword(s):  
Sufficient Conditions ◽  
Infinite Variance ◽  
Regularity Conditions ◽  
Self Similarity ◽  
Similarity Parameter ◽  
Stable Motion ◽  
Linear Fractional Stable Motion ◽  
Motion Process ◽  
Self Similar ◽  
Fractional Stable Motion

We study a family of locally self-similar stochastic processes Y = {Y(t)} t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

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