CHANCE AND STABILITY: STABLE DISTRIBUTIONS AND THEIR APPLICATIONS
CONTENTS I THEORY PROBABILITY PROBABILITY SPACE RANDOM VARIABLES FUNCTIONS X(?) RANDOM VECTORS AND PROCESSES INDEPENDENCE MEAN AND VARIANCE BERNOULLI THEOREM THE MOIVRE–LAPLACE THEOREM THE LAW OF LARGE NUMBERS 1 10 STRONG LAW OF LARGE NUMBERS 1 11 ERGODICITY AND STATIONARITY 1 12 THE CENTRAL LIMIT THEOREM ELEMENTARY INTRODUCTION TO THE THEORY OF STABLE LAWS CONVOLUTIONS OF DISTRIBUTIONS THE GAUSS DISTRIBUTION AND THE STABILITY PROPERTY THE CAUCHY AND L´EVY DISTRIBUTIONS SUMMATION OF STRICTLY STABLE RANDOM VARIABLES THE STABLE LAWS AS LIMITING DISTRIBUTIONS SUMMARY CHARACTERISTIC FUNCTIONS THE CHARACTERISTIC FUNCTIONS OF SYMMETRIC STABLE DISTRIBUTIONS SKEW STABLE DISTRIBUTIONS WITH ? < THE GENERAL FORM OF STABLE CHARACTERISTIC FUNCTIONS STABLE LAWS AS IN?NITELY DIVISIBLE LAWS VARIOUS FORMS OF STABLE CHARACTERISTIC FUNCTIONS SOME PROPERTIES OF STABLE RANDOM VARIABLES CONCLUSION PROBABILITY DENSITIES SYMMETRIC DISTRIBUTIONS CONVERGENT SERIES FOR ASYMMETRIC DISTRIBUTIONS LONG TAILS INTEGRAL REPRESENTATION OF STABLE DENSITIES INTEGRAL REPRESENTATION OF STABLE DISTRIBUTION FUNCTIONS DUALITY LAW SHORT TAILS STABLE DISTRIBUTIONS WITH ? CLOSE TO EXTREME VALUES INTEGRAL TRANSFORMATIONS LAPLACE TRANSFORMATION INVERSION OF THE LAPLACE TRANSFORMATION TAUBERIAN THEOREMS ONE-SIDED STABLE DISTRIBUTIONS LAPLACE TRANSFORMATION OF TWO-SIDED DISTRIBUTIONS THE MELLIN TRANSFORMATION THE CHARACTERISTIC TRANSFORMATION THE LOGARITHMIC MOMENTS MULTIPLICATION AND DIVISION THEOREMS SPECIAL FUNCTIONS AND EQUATIONS INTEGRODIFFERENTIAL EQUATIONS THE LAPLACE EQUATION FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS SPLITTING OF THE DIFFERENTIAL EQUATIONS SOME SPECIAL CASES THE WHITTAKER FUNCTIONS GENERALIZED INCOMPLETE HYPERGEOMETRICAL FUNCTION THE MEIJER AND FOX FUNCTIONS STABLE DENSITIES AS A CLASS OF SPECIAL FUNCTIONS 6 10 TRANSSTABLE FUNCTIONS 6 11 CONCLUDING REMARKS MULTIVARIATE STABLE LAWS BIVARIATE STABLE DISTRIBUTIONS TRIVARIATE STABLE DISTRIBUTIONS MULTIVARIATE STABLE DISTRIBUTIONS SPHERICALLY SYMMETRIC MULTIVARIATE DISTRIBUTIONS SPHERICALLY SYMMETRIC STABLE DISTRIBUTIONS SIMULATION THE INVERSE FUNCTION METHOD THE GENERAL FORMULA APPROXIMATE ALGORITHM FOR ONE-DIMENSIONAL SYMMETRIC STABLE VARIABLES SIMULATION OF THREE-DIMENSIONAL SPHERICALLY SYMMETRIC STABLE VECTORS ESTIMATION SAMPLE FRACTILE TECHNIQUE METHOD OF CHARACTERISTIC FUNCTIONS METHOD OF CHARACTERISTIC TRANSFORMS: ESTIMATORS OF ?, ? AND ? INVARIANT ESTIMATION OF ? ESTIMATORS OF PARAMETER ? MAXIMUM LIKELIHOOD ESTIMATORS FISHER’S INFORMATION FOR ? CLOSE TO 2 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 PRO?LE 13 2 STARK EFFECT IN AN ELECTRICAL ?ELD 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 10QUANTUM DECAY THEORY 13 11LOCALIZED VIBRATIONAL STATES (FRACTONS) 13 12ANOMALOUS TRANSIT-TIME IN SOME SOLIDS 13 13LATTICE PERCOLATION 13 14WAVES IN MEDIUM WITH MEMORY 13 15THE MESOSCOPIC EFFECT 13 16MULTIPARTICLE PRODUCTION 13 17TSALLIS’ DISTRIBUTIONS 13 18STABLE 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