References
- Beghin , L. , Orsingher , E. ( 2010 ). Poisson-type processes governed by fractional and higher-order recursive differential equations . Electro. J. Prob. 15 ( 22 ): 684 – 709 .
- Beghin , L. , Orsingher , E. ( 2009 ). Fractional Poisson processes and related planar random motions . Electro. J. Probab. 14 : 1790 – 1826 .
- Cahoy , D. O. ( 2007 ). Fractional Poisson Process in Terms of Alpha-Stable Densities. PhD Thesis, Case Western Reserve University .
- Cahoy , D. O. , Uchaikin , V. V. , Woyczynski , W. A. ( 2010 ). Parameter estimation in fractional Poisson processes . J. Statist. Plann. Infer. 140 ( 11 ): 3106 – 3120 .
- Germano , G. , Politi , M. , Scalas , E. , Schilling , R. L. ( 2009 ). Stochastic calculus for uncoupled continuous-time random walks. Phys. Rev. E 79:066102. http://pre.aps.org/abstract/ PRE/v79/i6/e066102 and http://arxiv.org/abs/ 0802.3769 .
- Gorenflo , R. , Luchko , Y. , Mainardi , F. ( 1999 ). Analytical properties and applications of the Wright function . Fract. Calculus Appl. Anal. 2 ( 4 ): 383 – 414 .
- Gorenflo , R. , Mainardi , F. ( 2009 ). Some recent advances in theory and simulation of fractional diffusion processes . J. Computat. Appl. Math. 229 ( 2 ): 400 – 415 .
- Haan , L. D. , Resnick , S. I. ( 1980 ). A simple asymptotic estimate for the index of a stable distribution . J. Roy. Statist. Soc. B 42 : 83 – 87 .
- Haubold, H. J., Mathai, A. M., Saxena, R. K. (2009). Mittag–Leffler functions and their applications. http://arxiv.org/abs/0909.0230
- Hill , B. M. ( 1975 ). A simple general approach to inference about the tail of a distribution . Ann. Statist. 3 : 1163 – 1174 .
- Jumarie , G. ( 2001 ). Fractional master equation: Non-standard analysis and Liouville–Riemann derivative . Chaos, Solitons Fractals 12 ( 13 ): 2577 – 2587 .
- Laskin , N. ( 2003 ). Fractional Poisson process . Commun. Nonlinear Sci. Num. Simul. 8 : 201 – 213 .
- Mainardi , F. , Gorenflo , R. , Scalas , E. ( 2003 ). Mellin transform and subordination laws in fractional diffusion processes. Fract. Calculus Appl. Anal. 6(4):441–459. E-print http://arxiv.org/abs/math/0702133 .
- Mainardi , F. , Gorenflo , R. , Scalas , E. ( 2004 ). A fractional generalization of the Poisson processes . Vietnam J. Math. 32 : 53 – 64 .
- Mainardi , F. , Luchko , Y. , Pagnini , G. ( 2001 ). The fundamental solution of the space-time fractional diffusion equation. Fract. Calculus Appl. Anal. 4(2):153–192. E-print http://arxiv.org/abs/cond-mat/0702419 .
- Mainardi , F. , Pagnini , G. ( 2003 ). The Wright functions as solutions of the time-fractional diffusion equation . Appl. Math. Comput. 141 : 51 – 62 .
- Mainardi , F. , Mura , A. , Pagnini , G. ( 2010 ). The M-Wright function in time-fractional diffusion processes: A tutorial survey. Int. J. Different. Eq. 2010. doi:10.1155/ 2010/104505 .
- Mainardi , F. , Tomirotti , M. (1997). Seismic pulse propagation with constant Q and stable probability distributions. Annali di Geofisica 40:1311–1328. E-print: http://arxiv.org/ abs/1008.1341.
- Mura , A. , Pagnini , G. ( 2008 ). Characterizations and simulations of a class of stochastic process to model anomalous diffusion. J. Phys. A Mathemat. Theoret. 41(28) .
- Mura, A., Taqqu, M. S., Mainardi, F. (2008). Non-Markovian diffusion equations and processes: Analysis and simulations. Physica A 387:5033–5064. E-print http:// arxiv.org/abs/0712.0240
- Podlubny , I. ( 1999 ). Fractional Differential Equations . San Diego : Academic Press .
- Repin , O. N. , Saichev , A. I. ( 2000 ). Fractional Poisson law . Radiophysics and Quantum Electronics 43 ( 9 ): 738 – 741 .
- Uchaikin , V. V. , Cahoy , D. O. , Sibatov , R. T. ( 2008 ). Fractional processes: From Poisson to branching one . Int. J. Bifurcation and Chaos 18 ( 9 ): 2717 – 2725 .
- Zolotarev , V. M. ( 1986 ). One-Dimensional Stable Distributions: Translations of Mathematical Monographs. Vol. 65. American Mathematical Society .