Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond

References

• Baleanu, D., K. Diethelm, E. Scalas, and J. J. Trujillo. 2012. Fractional calculus: Models and numerical methods. In Series on Complexity, Nonlinearity and Chaos Book Vol. 5, 2nd ed. Albemarle, NC: WSPC.
   Google Scholar
• Beghin, L., and E. Orsingher. 2009. Fractional Poisson processes and related planar random motions. Electronic Journal of Probability 14:1790–827. doi: 10.1214/EJP.v14-675.
   Web of Science ®Google Scholar
• Beghin, L., and E. Orsingher. 2010. Poisson-type processes governed by fractional and higher-order recursive differential equations. Electronic Journal of Probability 15:684–709. doi: 10.1214/EJP.v15-762.
   Web of Science ®Google Scholar
• Biard, R., and B. Saussereau. 2014. Fractional Poisson process: Long-range dependence and applications in ruin theory. Journal of Applied Probability 51 (3):727–40. Correction: 2016. J. Appl. Prob. 53: 1271–1272. doi: 10.1017/jpr.2016.80.
   Web of Science ®Google Scholar
• Chukova, S., and L. Minkova. 2015. Pólya-Aeppli of order k risk model. Communications in Statistics - Simulation and Computation 44 (3):551–64. doi: 10.1080/03610918.2013.784987.
   Web of Science ®Google Scholar
• Chukova, S., and L. Minkova. 2019. Non-homogeneous Pólya-Aeppli process. Communications in Statistics - Simulation and Computation 48 (10):2955–67. doi: 10.1080/03610918.2018.1469763.
   Web of Science ®Google Scholar
• Gupta, N., and A. Kumar. 2021. Fractional Poisson processes of order k and beyond. arXiv:2008.06022 [Math.PR].
   Google Scholar
• Gupta, N., A. Kumar, and N. Leonenko. 2020. Tempered fractional Poisson processes and fractional equations with Z-transform. Stochastic Analysis and Applications 38 (5):939–57. doi: 10.1080/07362994.2020.1748056.
   Web of Science ®Google Scholar
• Kataria, K. K., and P. Vellaisamy. 2018. On densities of the product, quotient and power of independent subordinators. Journal of Mathematical Analysis and Applications. 462 (2):1627–43. doi: 10.1016/j.jmaa.2018.02.059.
   Web of Science ®Google Scholar
• Kostadinova, K., and M. Lazarova. 2019. Risk models of order k. Ann. Acad. Rom. Sci. Ser. Math. Appl 11 (2):259–73.
   Google Scholar
• Kostadinova, K., and L. Minkova. 2019. On the Poisson process of order k. Pliska Stud. Math. Bulgar 22:117–28.
   Google Scholar
• Kumar, A., N. Leonenko, and A. Pichler. 2020. Fractional risk process in insurance. Mathematics and Financial Economics 14 (1):43–65. doi: 10.1007/s11579-019-00244-y.
   Web of Science ®Google Scholar
• Leonenko, N., M. M. Meerschaert, R. Schilling, and A. Sikorskii. 2014. Correlation structure of time changed Lévy proceses. Communications in Applied and Industrial Mathematics 6 (1):483–505.
   Web of Science ®Google Scholar
• Leonenko, N., E. Scalas, and M. Trinh. 2017. The fractional non-homogeneous Poisson process. Statistics & Probability Letters 120:147–56. doi: 10.1016/j.spl.2016.09.024.
   Web of Science ®Google Scholar
• Mainardi, F., R. Gorenflo, and E. Scalas. 2004. A fractional generalization of the Poisson processes. Vietnam Journal of Mathematics 32:53–64.
   Google Scholar
• Meerschaert, M. M., E. Nane, and P. Vellaisamy. 2011. Fractional Poisson process and the inverse stable subordinator. Electronic Journal of Probability 16 (none):1600–20. doi: 10.1214/EJP.v16-920.
   Web of Science ®Google Scholar
• Meerschaert, M. M., and A. Sikorskii. 2019. Stochastic Models For Fractional Calculus. De Gruyter Studies in Mathematics, Vol. 43, 2nd ed. Berlin: Walter de Gruyter.
   Google Scholar
• Minkova, L. D. 2010. The Pólya-Aeppli distribution of order k. Communications in Statistics - Theory and Methods 39 (3):408–15. doi: 10.1080/03610920903140072.
   Web of Science ®Google Scholar
• Orsingher, E., and F. Polito. 2010. Fractional pure birth processes. Bernoulli 16 (3):858–81. doi: 10.3150/09-BEJ235.
   Web of Science ®Google Scholar
• Philippou, A. N. 1983. The Poisson and compound Poisson distribution of order k and their properties. Zapiski Nauchnyh Seminarov Instituta im. Steklova 130:175–80.
   Google Scholar
• Philippou, A. N., C. Georghiou, and G. N. Philippou. 1983. A generalized geometric distribution and some of its properties. Statistics & Probability Letters 1 (4):171–5. doi: 10.1016/0167-7152(83)90025-1.
   Google Scholar
• Sengar, A. S., A. Maheshwari, and N. S. Upadhye. 2020. Time-changed Poisson process of order k. Stochastic Analysis and Applications 38 (1):124–48. doi: 10.1080/07362994.2019.1653198.
   Web of Science ®Google Scholar