This page is a collection of references related to the theory and applications of alpha-stable processes. It is by no means complete and is focused on references that I use in my own work. For a more comprehensive list, see Nolan’s collection.

**Standard References**

Samorodnitsky, G. and Taqqu, M., *Stable Non-Gaussian Random Processes*. Chapman and Hall, 1994.

Uchaikin, V.V. and Zolotarev, V.M., *Chance and Stability: Stable Distributions and Their Applications. *Walter de Gruyter, 1999.

Zolotarev, V.M., *One-Dimensional Stable Distributions. *American Mathematical Society, 1986.

**In One-Dimension**

Zolotarev, V.M., “Mellin-Stieltjes transforms in probability theory,” *Theory Probab. Appl., *vol. 2, no. 4, 1957.

Fofack, H. and Nolan, J., “Tail behavior, modes and other characteristics of stable distributions,” *Extremes*, vol. 2, no. 1, pp. 39-58, 1999.

Koutrouvelis, I.A., “Regression-type estimation of the parameters of stable laws,” *Journal of the American Statistical Association*, vol. 75, no. 372, 1980.

Feuerverger, A. and Mureika, R.A., “The empirical characteristic function and its applications,” *The Annals of Statistics*, vol. 5, no. 1, pp. 88-97, 1977.

Cottone, G. and Di Paola, M., “On the use of fractional calculus for the probabilistic characterization of random variables,” *Probabilistic Engineering Mechanics, *vol. 23, no. 3, pp. 321-330, 2009.

Di Paola, M., Pinnola, F.P., “Riesz fractional integrals and complex fractional moments for the probabilistic characterization of random variables,” *Probabilistic Engineering Mechanics*, vol. 29, no. 3, 2012.

Menn, C. and Rachev, S.T., “Calibrated FFT-based density approximations for alpha-stable distributions,” *Computational Statistics and Data Analysis, *vol. 50, no. 8, pp. 1891-1904, 2006.

Nolan, J.P., “An algorithm for evaluating stable densities in Zolotarev’s (M) Parameterization,” *Mathematical and Computer Modelling, *vol. 29, pp. 229-233, 1999.

**Random Vectors**

Nolan, J.P., Panorska, A.K. and McCulloch, J.H., “Estimation of stable spectral measures,” *Mathematical and Computer Modelling*, vol. 34, pp. 1113-1122, 2001.

Nolan, J.P. “Multivariate elliptically contoured stable distributions: theory and estimation,” *Computational Statistics*, vol. 28, no. 5, pp. 2067-2089, 2013.

**Processes via Spectral Representations**

Cambanis, S. and Miller, G., *“*Linear problems in p-th order and symmetric stable processes,” *SIAM Journal of Applied Mathematics*, vol. 41, pp 43-69, 1981.

Rao, M.M., “Harmonizable processes: structure theory,” *L’enseignement Mathématiques**, *vol. 28, 1982.

Rao, M.M., “Harmonizable, Cramer, and Karhunen classes of processes,” *DTIC Technical Report*, 1984.

Li, K.-S. and Rosenblatt, M., “Spectral analysis for harmonizable processes,” *Annals of Statistics, *vol. 30, pp. 438-442, 2002.

**Related Theory**

Morse, M. and Transue, W., “C-bimeasures,” *The Annals of Mathematics*, vol. 64, pp. 480-504, 1956.

Samko, S.G., Kilbas, A.A. and Marichev, O.I., *Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach Science Publishers, Amsterdam, 1993.

**Applications to Communication Theory**

Azzaoui, N., Clavier, L., Guillin, A. and Peters, G.W., *Spectral Measures of Alpha-Stable Distributions: An Overview and Natural Applications in Wireless* *Communications*. * *Springer Briefs, Japan, 2015.

Nikias, C.L. and Shao, M., *Signal Processing with alpha-stable distributions and applications*. Wiley-Interscience, 1995.

Pinto, P. and Win, M., “Communication in a Poisson field of interferers-part I: interference distribution and error probability,” *IEEE Transactions on Wireless Communications*, vol. 9, no. 7, pp. 2176-2186, 2010.

Pinto, P. and Win, M., “Communication in a Poisson field of interferers-part II: channel capacity and interference spectrum,” *IEEE Transactions on Wireless Communications*, vol. 9, no. 7, pp. 2187-2195, 2010.

Nikfar, B., Akbudak, T. and Han Vinck, A., “MIMO capacity of class A impulsive noise channel for different levels of information availability at transmitter,” in *Proc. IEEE International Symposium on Power Line Communications and Its Applications*, 2014.

Zha, D. and Qui, T., “Underwater sources location in non-Gaussian impulsive noise environments,” *Digital Signal Processing, *vol. 16, pp. 149-163, 2006.

Farsad, N., Guo, W., Chae, C.-B. and Eckford, A., “Stable distributions as noise models for molecular communication,” in *Proc. IEEE Global Communications Conference*, 2015.

Azzaoui, N. and Clavier, L., “Statistical channel model based on alpha-stable random processes and application to the 60 GHz ultra wide band channel,” *IEEE Transactions on Communications*, vol. 58, no. 5, pp. 1457-1467, 2010.

El Ghannudi, H., Clavier, L., Azzaoui, N., Septier, F. and Rolland, P.A., “Alpha-stable interference modeling and Cauchy receiver for an IR-UWB ad hoc network,” *IEEE Transactions on Communications, *vol. 58, no. 6, pp. 1748-1757, 2010.

Yang, X. and Petropulu, A.P., “Co-channel interference modeling and analysis in a Poisson field of interferers in wireless communications,” *IEEE Transactions on Signal Processing*, vol. 51, no. 1, pp. 64-76, 2003.

Gulati, K., Chopra, A., Evans, B.L. and Tinsley, K.R., “Statistical modeling of co-channel interference,” in *Proc. IEEE Global Telecommunications Conference*, 2009.

Gulati, K., Evans, B.L., Andrews, J.G. and Tinsley, K.R., “Statistics of co-channel interference in a field of Poisson and Poisson-Poisson clustered interferers,” *IEEE Transactions on Signal Processing*, vol. 58, no. 12, pp. 6207-6222, 2010.

**Applications to Information Theory**

Egan, M., de Freitas, M., Clavier, L., Peters, G.W. and Azzaoui, N., “Achievable rates for additive alpha-stable noise channels,” in *Proc. IEEE International Symposium on Information Theory (ISIT), *2016.

Fahs, J. and Abou-Faycal, I., “On the finiteness of the capacity of continuous channels,” *IEEE Transactions on Communication,* vol. 54, no. 1, pp. 166-173, 2016.

Fahs, J. and Abou-Faycal, I., “A Cauchy input achieves the capacity of a Cauchy channel under a logarithmic constraint,” in *Proc. IEEE International Symposium on Information Theory (ISIT)*, 2014.

Fahs, J. and Abou-Faycal, I., “On the capacity of additive white alpha-stable noise channels,” in *Proc. IEEE International Symposium on Information Theory (ISIT)*, 2012.

Fahs, J. and Abou-Faycal, I., “Input constraints and noise density functions: a simple relation for bounded-support and discrete capacity-achieving inputs,” *arXiv:1602.00878*, 2016.

**Software**

Veillette, M., “Alpha-stable distributions,” available: http://math.bu.edu/people/mveillet/html/alphastablepub.html