Chance and Stability: Stable Distributions and their Applications

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Walter de Gruyter, Sep 8, 2011 - Mathematics - 598 pages

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This book is the only chance you have to understand the generalization of the central limit theorem for stable distributions

Contents

112 Markov point distributions
301
113 Average density of random distribution
303
114 Correlation functions
306
115 Inverse power type correlations and stable distributions
310
116 Mandelbrots stochastic fractals
315
117 Numerical results
319
118 Fractal sets with a turnover to homogeneity
322
12 Anomalous diffusion and chaos
331

19 The law of large numbers
22
110 Strong law of large numbers
24
111 Ergodicity and stationarity
28
112 The central limit theorem
31
2 Elementary introduction to the theory of stable laws
35
22 The Gauss distribution and the stability property
39
23 The Cauchy and LÚvy distributions
44
24 Summation of strictly stable random variables
50
25 The stable laws as limiting distributions
55
26 Summary
64
3 Characteristic functions
69
32 The characteristic functions of symmetric stable distributions
72
33 Skew stable distributions with α 1
77
34 The general form of stable characteristic functions
82
35 Stable laws as infinitely divisible laws
86
36 Various forms of stable characteristic functions
93
37 Some properties of stable random variables
98
38 Conclusion
102
4 Probability densities
103
42 Convergent series for asymmetric distributions
105
43 Long tails
110
44 Integral representation of stable densities
115
45 Integral representation of stable distribution functions
119
46 Duality law
121
47 Short tails
123
48 Stable distributions with α close to extreme values
127
49 Summary
130
5 Integral transformations
137
52 Inversion of the Laplace transformation
140
53 Tauberian theorems
143
54 Onesided stable distributions
145
55 Laplace transformation of twosided distributions
150
56 The Mellin transformation
152
57 The characteristic transformation
154
58 The logarithmic moments
156
59 Multiplication and division theorems
158
6 Special functions and equations
167
62 The Laplace equation
169
63 Fractional integrodifferential equations
171
64 Splitting of the differential equations
174
65 Some special cases
176
66 The Whittaker functions
178
67 Generalized incomplete hypergeometrical function
179
68 The Meijer and Fox functions
181
69 Stable densities as a class of special functions
185
610 Transstable functions
188
611 Concluding remarks
190
7 Multivariate stable laws
193
72 Trivariate stable distributions
200
73 Multivariate stable distributions
203
74 Spherically symmetric multivariate distributions
207
75 Spherically symmetric stable distributions
210
8 Simulation
213
82 The general formula
216
83 Approximate algorithm for onedimensional symmetric stable variables
218
84 Simulation of threedimensional spherically symmetric stable vectors
221
9 Estimation
229
92 Method of characteristic functions
231
estimators of ν θ and τ
232
94 Invariant estimation of α
240
95 Estimators of parameter γ
241
96 Maximum likelihood estimators
245
97 Fishers information for a close to 2
249
98 Concluding remarks
251
II Applications
253
10 Some probabilistic models
255
102 Stable laws in games
257
103 Random walks and diffusion
261
104 Stable processes
265
105 Branching processes
274
twodimensional case
281
multidimensional case
286
108 A class of sources generating stable distributions
290
11 Correlated systems and fractals
297
122 Two examples of anomalous diffusion
333
123 Superdiffusion
336
124 Subdiffusion
343
125 CTRW equations
349
126 Some special cases
352
127 Asymptotic solution of the MontrollWeiss problem
356
128 Twostate model
358
129 Stable laws in chaos
361
13 Physics
365
132 Stark effect in an electrical field of randomly distributed ions
367
133 Dipoles and quadrupoles
372
134 Landau distribution
373
135 Multiple scattering of charged particles
377
136 Fractal turbulence
381
137 Stresses in crystalline lattices
382
138 Scaleinvariant patterns in acicular martensites
383
139 Relaxation in glassy materials
384
1310 Quantum decay theory
386
1311 Localized vibrational states fractons
389
1312 Anomalous transittime in some solids
390
1313 Lattice percolation
392
1314 Waves in medium with memory
394
1315 The mesoscopic effect
397
1316 Multiparticle production
398
1317 Tsallis distributions
400
1318 Stable distributions and renormalization group
402
14 Radiophysics
405
142 Distortion of information phase
407
143 Signal and noise in a multichannel system
410
144 Wave scattering in turbulent medium
413
145 Chaotic phase screen
416
15 Astrophysics and cosmology
419
152 Cosmic rays
420
153 Stellar dynamics
424
154 Cosmological monopole and dipole
428
155 The Universe as a rippled water
429
156 The power spectrum analysis
431
157 Cellcount distribution for the fractal Universe
434
158 Global mass density for the fractal Universe
435
16 Stochastic algorithms
439
162 Flux at a point
441
163 Examples
443
164 Estimation of a linear functional of a solution of integral equation
445
165 Random matrices
453
166 Random symmetric polynomials
454
17 Financial applications
463
172 More on stable processes
464
173 Multivariate stable processes
466
174 Stable portfolio theory
469
175 Logstable option pricing
474
176 Low probability and shortlived options
481
177 Parameter estimation and empirical issues
482
18 Miscellany
489
182 Genetics
493
183 Physiology
496
184 Ecology
498
185 Geology
499
Appendix
501
A1 Onedimensional densities qAx α β
503
A2 Onesided distribution functions GBx α 1 multiplied by 104
510
A3 Onesided distributions represented by function Fy α 104GBy1α α 1 Fy 0 104ey
511
A4 The function α1α qα1α x α where qx α is the onedimensional symmetric stable density
513
A5 Radial functions ρ2r α of twodimensional axially symmetric densities
514
A6 Radial functions ρ3r α of threedimensional spherically symmetric densities
515
A7 Strictly stable densities expressed via elementary functions special functions and quadratures
516
A8 Fractional integrodifferential operators
518
A9 Approximation of inverse distribution function rx F1x for simulation of threedimensional random vectors with density q3rα
522
A10 Some statistical terms
524
A11 Some auxiliary formulae for statistical estimators
525
A12 Functional derivatives
526
Conclusion
531
Bibliography
533
Index
569
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