References

Books

These are hopefully organized in order of usefulness for applications and understandability.

Nolan, J.P., Stable Distributions, Models for Heavy Tailed Data, Birkhauser, (?).

This is not yet published, but chapter 1 is available online and provides a good introduction to stable distributions. There is a great deal of other information on this site which is the best starting place for stable distributions.

C. Rose and M. D. Smith, Mathematical Statistics with Mathematica, Springer Texts in Statistics, New York: Springer-Verlag, 2002.

Bouchaud, M. and Potters, Theory Financial Risk and Derivative Pricing, Cambridge, 2003.

McNeil, A.J., Frey, R., Embrechts, P. Quantitative Risk Management, Concepts, Techniques, Tools, Princeton University Press 2005.

Adler,R., Feldman, R.,Taqqu, M.,(editors), A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Birkhauser, 1998.

Rachev, S., Fabozzi, F., Menn, C., Fat-Tailed and Skewed Asset Return Distributions : Implications for Risk Management, Portfolio Selection, and Option Pricing, Wiley, 2005.

Rachev, S. T., Handbook of Heavy Tailed Distributions in Finance, Elsevier. 2003.

S. T. Rachev and S. Mittnik, Stable Paretian Models in Finance, New York: John Wiley, 2000.

B. B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, New York: Springer-Verlag, 1997.

Feller, W., An Introduction to Probability Theory Vol. II, John Wiley & Sons. 1971.

Gnedenko, B.V., Kolmogorov, A.N., Limit Distributions for Sums of Independent Random Variables, Addison-Wesley 1968.

Samorodnitsky, G. and M. Taqqu. Stable Non-Gaussian Random Processes,

New York: Chapman and Hall. 1994.

V. M. Zolotarev, One-Dimensional Stable Distributions, Translation of Mathematical Monographs, Vol. 65, American Mathematical Society, 1986. (Translation of the original 1983 Russian).

Papers

John Nolan maintains an extensive Stable Bibliography which is on line: StableBibliography.pdf.

A few other papers that may be referenced on the web site or are classical are below.

Mandelbrot, B. (1963). The Variation of Certain Speculative Prices, J.Business, 36, 394-419.

The article is reprinted with comments in:

Mandelbrot, B.(1997). Fractals and Scaling in Finance, Discontinuity Concentration, Risk, Springer, 1997.

Plerou, V., Gopikrishnan, P., Amaral, L., Meyer, M., Stanley, H.E. (1999) Scaling of the distribution of price fluctuations of individual companies. Physical Review E, 60:6

Gabaix, X, Gopikrishnan, P., Plerou, V., Stanley E. (2007) A unified econophysics explanation for the power-law exponents of stock market activity. Physica A 382:81–88

McCulloch, J.H. (1997). Measuring Tail Thickness to Estimate the Stable Index α: A Critique, J. Business & Economic Statistics 15, 74-81.

Smith, E., Farmer, J.D., Gillemot, L., Krishnamurthy, S., Statistical theory of the continuous double auction, Quantitative Finance, 3, 481-514. Farmer's web site has many useful references.

Rimmer, R., Nolan, J.P.(2005) Stable Distributions in Mathematica. The Mathematica Journal 9(4) : 776-789.

This paper describes an earlier version of StableM.

J. P. Nolan, Numerical Calculation of Stable Densities and Distribution Functions, Communications in Statistics-Stochastic Models, 13(4), 1997 pp. 759–774.

J. M. Chambers, C. L. Mallows, and B. W. Stuck, A Method for Simulating Stable Random Variables, Journal of the American Statistical Association, 71(354), 1976 pp. 340–344.

Reed, W. J. The normal-Laplace distribution and its relatives. To appear Birkhauser. This and a number of papers on the the Pareto - log normal distribution are available on Dr. Reed's website. These distributions will likelly become important in the structure of limit order books and they provide an explanation for the evolution of power tail distribuitons in finance. Sums of power tail random variables have as their limit stable distributions when the tail exponent is less than two.

Limpert, E., Stahel, W., Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues." BioScience 51: 5 May (2001) p. 341. Available at: http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

Mathematica Software

J. P. Nolan, Stable MathLink Package, www.robustanalysis.com.

R. H. Rimmer, StableM7. mathestate software page.

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