Stable distributions

We give many explicit formulas for stable distributions, mainly based on Feller [3] and Zolotarev [14] and using several parametrizations; we give also some explicit calculations for convergence to stable distributions, mainly based on less explicit results in Feller [3]. The main purpose is to provide ourselves with easy reference to explicit formulas and examples. (There are probably no new results.)

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  B. Mandlebrot.  The Variation of Certain Speculative Prices , 1963 .

[3]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[4]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[5]  P. Hall.  ONE‐DIMENSIONAL STABLE DISTRIBUTIONS (Translations of Mathematical Monographs 65) , 1987 .

[6]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[7]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[8]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[9]  S. Rachev,et al.  Stable Paretian Models in Finance , 2000 .

[10]  R. Wolpert.  Lévy Processes , 2000 .

[11]  J. L. Nolan.  Stable Distributions. Models for Heavy Tailed Data , 2001 .

[12]  Philippe Flajolet,et al.  Random maps, coalescing saddles, singularity analysis, and Airy phenomena , 2001, Random Struct. Algorithms.

[13]  A. ADoefaa,et al.  ? ? ? ? f ? ? ? ? ? , 2003 .

[14]  Pukelsheim Friedrich,et al.  Sainte-Laguë’s chi-square divergence for the rounding of probabilities and its convergence to a stable law , 2004 .

[15]  S. Rachev,et al.  A New Class of Probability Distributions and its Application to Finance , 2004 .

[16]  Pukelsheim Friedrich,et al.  On stationary multiplier methods for the rounding of probabilities and the limiting law of the Sainte-Laguë divergence , 2005 .

[17]  Svante Janson,et al.  Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation , 2011, 1112.0510.