## Stable Distributions - Semantic Scholar

4.12 Fitting stable distributions to concentration data . . . . . . . . . . . . . . . 182 .... III Regression, Time Series, Signal Processing and Stable Processes 309. 9 Stable ..... We now have computer programs to compute quantities of interest for stable ...
Stable Distributions Models for Heavy Tailed Data John P. Nolan [email protected] Math/Stat Department American University c Copyright ⃝2015 John P. Nolan Processed February 14, 2016

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Contents

I Univariate Stable Distributions 1 Basic Properties of Univariate Stable Distributions 1.1 Definition of stable . . . . . . . . . . . . . . . 1.2 Other definitions of stability . . . . . . . . . . 1.3 Parameterizations of stable laws . . . . . . . . 1.4 Densities and distribution functions . . . . . . 1.5 Tail probabilities, moments and quantiles . . . 1.6 Sums of stable random variables . . . . . . . . 1.7 Simulation . . . . . . . . . . . . . . . . . . . . 1.8 Generalized Central Limit Theorem . . . . . . 1.9 Problems . . . . . . . . . . . . . . . . . . . .

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2 Modeling with Stable Distributions 2.1 Lighthouse problem . . . . . . . . . . . . . . . . . . . . . . . 2.2 Distribution of masses in space . . . . . . . . . . . . . . . . . 2.3 Random walks . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hitting time for Brownian motion . . . . . . . . . . . . . . . 2.5 Differential equations and fractional diffusions . . . . . . . . 2.6 Economic applications . . . . . . . . . . . . . . . . . . . . . 2.6.1 Stock returns . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Foreign exchange rates . . . . . . . . . . . . . . . . . 2.6.3 Value-at-risk . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Other economic applications . . . . . . . . . . . . . . 2.6.5 Long tails in business, political science, and medicine

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3 Technical Results on Univariate Stable Distributions 3.1 Proofs of Basic Theorems of Chapter 1 . . . . . . . . . . . . 3.1.1 Stable distributions as infinitely divisible distributions 3.2 Densities and distribution functions . . . . . . . . . . . . . . 3.2.1 Series expansions . . . . . . . . . . . . . . . . . . . . 3.2.2 Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical algorithms . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Computation of distribution functions and densities . . 3.3.2 Spline approximation of densities . . . . . . . . . . . 3.3.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . 3.4 More on parameterizations . . . . . . . . . . . . . . . . . . . 3.5 Tail behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Moments and other transforms . . . . . . . . . . . . . . . . . 3.7 Convergence of stable laws in terms of (α , β , γ , δ ) . . . . . . . 3.8 Combinations of stable random variables . . . . . . . . . . . . 3.9 Distributions derived from stable distributions . . . . . .