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