Chance and Stability: Stable Distributions and their Applications

✍ Scribed by Vladimir V. Uchaikin; Vladimir M. Zolotarev

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 600
• Category: Library

✦ Table of Contents

Foreword
Introduction
I Theory
1 Probability
1.1 Probability space
1.2 Random variables
1.3 Functions Χ (λ)
1.4 Random vectors and processes
1.5 Independence
1.6 Mean and variance
1.7 Bernoulli theorem
1.8 The Moivre-Laplace theorem
1.9 The law of large numbers
1.10 Strong law of large numbers
1.11 Ergodicity and stationarity
1.12 The central limit theorem
2 Elementary introduction to the theory of stable laws
2.1 Convolutions of distributions
2.2 The Gauss distribution and the stability property
2.3 The Cauchy and Lévy distributions
2.4 Summation of strictly stable random variables
2.5 The stable laws as limiting distributions
2.6 Summary
3 Characteristic functions
3.1 Characteristic functions
3.2 The characteristic functions of symmetric stable distributions
3.3 Skew stable distributions with α < 1
3.4 The general form of stable characteristic functions
3.5 Stable laws as infinitely divisible laws
3.6 Various forms of stable characteristic functions
3.7 Some properties of stable random variables
3.8 Conclusion
4 Probability densities
4.1 Symmetric distributions
4.2 Convergent series for asymmetric distributions
4.3 Long tails
4.4 Integral representation of stable densities
4.5 Integral representation of stable distribution functions
4.6 Duality law
4.7 Short tails
4.8 Stable distributions with α close to extreme values
4.9 Summary
5 Integral transformations
5.1 Laplace transformation
5.2 Inversion of the Laplace transformation
5.3 Tauberian theorems
5.4 One-sided stable distributions
5.5 Laplace transformation of two-sided distributions
5.6 The Mellin transformation
5.7 The characteristic transformation
5.8 The logarithmic moments
5.9 Multiplication and division theorems
6 Special functions and equations
6.1 Integrodifferential equations
6.2 The Laplace equation
6.3 Fractional integrodifferential equations
6.4 Splitting of the differential equations
6.5 Some special cases
6.6 The Whittaker functions
6.7 Generalized incomplete hypergeometrical function
6.8 The Meijer and Fox functions
6.9 Stable densities as a class of special functions
6.10 Transstable functions
6.11 Concluding remarks
7 Multivariate stable laws
7.1 Bivariate stable distributions
7.2 Trivariate stable distributions
7.3 Multivariate stable distributions
7.4 Spherically symmetric multivariate distributions
7.5 Spherically symmetric stable distributions
8 Simulation
8.1 The inverse function method
8.2 The general formula
8.3 Approximate algorithm for one-dimensional symmetric stable variables
8.4 Simulation of three-dimensional spherically symmetric stable vectors
9 Estimation
9.1 Sample fractile technique
9.2 Method of characteristic functions
9.3 Method of characteristic transforms: estimators of ν, θ and τ
9.4 Invariant estimation of α
9.5 Estimators of parameter γ
9.6 Maximum likelihood estimators
9.7 Fisher’s information for a close to 2
9.8 Concluding remarks
II Applications
10 Some probabilistic models
10.1 Generating functions
10.2 Stable laws in games
10.3 Random walks and diffusion
10.4 Stable processes
10.5 Branching processes
10.6 Point sources: two-dimensional case
10.7 Point sources: multidimensional case
10.8 A class of sources generating stable distributions
11 Correlated systems and fractals
11.1 Random point distributions and generating functionals
11.2 Markov point distributions
11.3 Average density of random distribution
11.4 Correlation functions
11.5 Inverse power type correlations and stable distributions
11.6 Mandelbrot’s stochastic fractals
11.7 Numerical results
11.8 Fractal sets with a turnover to homogeneity
12 Anomalous diffusion and chaos
12.1 Introduction
12.2 Two examples of anomalous diffusion
12.3 Superdiffusion
12.4 Subdiffusion
12.5 CTRW equations
12.6 Some special cases
12.7 Asymptotic solution of the Montroll-Weiss problem
12.8 Two-state model
12.9 Stable laws in chaos
13 Physics
13.1 Lorentz dispersion profile
13.2 Stark effect in an electrical field of randomly distributed ions
13.3 Dipoles and quadrupoles
13.4 Landau distribution
13.5 Multiple scattering of charged particles
13.6 Fractal turbulence
13.7 Stresses in crystalline lattices
13.8 Scale-invariant patterns in acicular martensites
13.9 Relaxation in glassy materials
13.10 Quantum decay theory
13.11 Localized vibrational states (fractons)
13.12 Anomalous transit-time in some solids
13.13 Lattice percolation
13.14 Waves in medium with memory
13.15 The mesoscopic effect
13.16 Multiparticle production
13.17 Tsallis’ distributions
13.18 Stable distributions and renormalization group
14 Radiophysics
14.1 Transmission line
14.2 Distortion of information phase
14.3 Signal and noise in a multichannel system
14.4 Wave scattering in turbulent medium
14.5 Chaotic phase screen
15 Astrophysics and cosmology
15.1 Light of a distant star
15.2 Cosmic rays
15.3 Stellar dynamics
15.4 Cosmological monopole and dipole
15.5 The Universe as a rippled water
15.6 The power spectrum analysis
15.7 Cell-count distribution for the fractal Universe
15.8 Global mass density for the fractal Universe
16 Stochastic algorithms
16.1 Monte-Carlo estimators with infinite variance
16.2 Flux at a point
16.3 Examples
16.4 Estimation of a linear functional of a solution of integral equation
16.5 Random matrices
16.6 Random symmetric polynomials
17 Financial applications
17.1 Introduction
17.2 More on stable processes
17.3 Multivariate stable processes
17.4 Stable portfolio theory
17.5 Log-stable option pricing
17.6 Low probability and short-lived options
17.7 Parameter estimation and empirical issues
18 Miscellany
18.1 Biology
18.2 Genetics
18.3 Physiology
18.4 Ecology
18.5 Geology
Appendix
A.1 One-dimensional densities qA(x; α, β)
A.2 One-sided distribution functions GB(x; α, 1) multiplied by 104
A.3 One-sided distributions represented by function F(y; α) = 104GB(y-1/α; α, 1) (F(y; 0) ≡ 104e-y)
A.4 The function α1/α q(α1/α x; α), where q(x; α) is the one-dimensional symmetric stable density
A.5 Radial functions ρ2(r; α) of two-dimensional axially symmetric densities
A.6 Radial functions ρ3(r; α) of three-dimensional spherically symmetric densities
A.7 Strictly stable densities expressed via elementary functions, special functions and quadratures
A.8 Fractional integro-differential operators
A.9 Approximation of inverse distribution function r(x) = F-1(x) for simulation of three-dimensional random vectors with density q3(r;α)
A.10 Some statistical terms
A.11 Some auxiliary formulae for statistical estimators
A.12 Functional derivatives
Conclusion
Bibliography
Index