In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller’s characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the Lévy-Wiener theorem. This is a special case of an open problem which is proposed by Sato (2014), Chaumont and Yor (2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter λ in the DPCP class cannot be arbitrarily small.
ProCite
RefWorks
Reference Manager
BibTeX
Zotero
EndNote
-
[1]
Albeverio, S., Gottschalk, H., Wu, J. L., Partly divisible probability distributions, Forum Mathematicum, 10(6) (1998), 687–698.
-
[2]
Baez-Duarte, L., Central limit theorem for complex measures, Journal of Theoretical Probability, 6(1) (1993), 33–56.
-
[3]
Baishanski, B., Norms of powers and a central limit theorem for complex-valued probabilities, in: Analysis of Divergence (pp. 523–543). Birkhäuser, Boston, 1998.
-
[4]
Beghin, L., Macci, C., Fractional discrete processes: compound and mixed Poisson representations, Journal of Applied Probability, 51 (2014), 19–36.
-
[5]
Beurling, A., Sur les intégrales de Fourier absolument convergentes et leur applicationa une transformation fonctionnelle, in: Ninth Scandinavian Mathematical Congress (pp. 345–366), 1938.
-
[6]
Beurling, A., Helson, H., Fourier-Stieltjes transforms with bounded powers, Mathematica Scandinavica, 1 (1953), 120–126.
-
[7]
Buchmann, B., GrÜbel, R., Decompounding: an estimation problem for Poisson random sums, The Annals of Statistics, 31(4) (2003), 1054–1074.
-
[8]
Bogachev, V. I., Measure theory (Vol. II.). Springer, Berlin, 2007.
-
[9]
Chaumont, L., Yor, M., Exercises in Probability: a guided tour from measure theory to random processes, via conditioning, Second edition, Cambridge University Press, 2012.
-
[10]
Demni, N., Mouayn, Z., Analysis of generalized Poisson distributions associated with higher Landau levels, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 18(04) (2015), 1550028.
-
[11]
Feller, W., An introduction to probability theory and its applications, Vol. I. 3ed., Wiley, New York.
-
[12]
Feller, W., An introduction to probability theory and its applications, Vol. II. 2ed., Wiley, New York, 2071.
-
[13]
Haight, F. A., Handbook of the Poisson distribution, Wiley, Los Angeles, 1967.
-
[14]
Hürlimann, W., Pseudo compound Poisson distributions in risk theory, ASTIN Bulletin, 20(1) (1990), 57–79.
-
[15]
Itô, K., Stochastic processes: Lectures given at Aarhus University, Springer, Berlin, 2004.
-
[16]
Jánossy, L., Rényi, A., Aczél, J., On composed Poisson distributions, I., Acta Mathematica Hungarica, 1(2) (1950), 209–224.
-
[17]
Johnson, N. L., Kemp, A. W., Kotz, S., Univariate Discrete Distributions, 3ed., Wiley, New Jersey, 2005.
-
[18]
Jørgensen, B., The theory of dispersion models, Chapman & Hall, London, 1997.
-
[19]
Karymov, D. N., On the decomposition of lattice distributions into convolutions of Poisson signed measures, Theory of Probability & Its Applications, 49(3) (2005), 545–552.
-
[20]
Kemp, A. W., Convolutions involving binomial pseudo-variables, Sankhya: The Indian Journal of Statistics, Series A (1979), 232–243.
-
[21]
Kerns, G. J., Signed measures in exchangeability and infinite divisibility (Doctoral dissertation, Bowling Green State University), 2004.
-
[22]
Laskin, N., Fractional Poisson process, Communications in Nonlinear Science and Numerical Simulation, 8(3) (2003), 201–213.
-
[23]
Letac, G., Malouche, D., Maurer, S., The real powers of the convolution of a negative binomial distribution and a Bernoulli distribution, Proceedings of the American Mathematical Society, 130(7) (2002), 2107–2114.
-
[24]
Lévy, P., Sur la convergence absolue des séries de Fourier, Compositio Mathematica, 1 (1935), 1–14.
-
[25]
Lévy, P., Théorie de l’addition des variables aléatoires, Gauthiers-Villars, Paris, 1937.
-
[26]
Lévy, P., Sur les exponentielles de polynômes et sur l’arithmétique des produits de lois de Poisson, Annales scientifiques de l’École Normale Supérieure, 54 (1937), 231–292.
-
[27]
Lindner, A., Pan, L., Sato, K., On quasi-infinitely divisible distributions, to appear in Trans. Amer. Math. Soc. https://arxiv.org/abs/1701.02400.
-
[28]
Maceda, E. C., On the compound and generalized Poisson distributions, The Annals of Mathematical Statistics, 19(3) (1948), 414–416.
-
[29]
Meerschaert, M. M., Scheffler, H. P., Limit distributions for sums of independent random vectors: Heavy tails in theory and practice, Wiley, New York, 2001.
-
[30]
Milne, R. K., Westcott, M., Generalized multivariate Hermite distributions and related point processes,. Annals of the Institute of Statistical Mathematics, 45(2) (1993), 367–381.
-
[31]
Nakamura, T., A quasi-infinitely divisible characteristic function and its exponentiation, Statistics & Probability Letters, 83(10) (2013), 2256–2259.
-
[32]
Nakamura, T., A complete Riemann zeta distribution and the Riemann hypothesis, Bernoulli, 21(1) (2015), 604–617.
-
[33]
Nahla, B. S., Afif, M., The real powers of the convolution of a gamma distribution and a Bernoulli distribution, Journal of Theoretical Probability, 24(2) (2011), 450–453.
-
[34]
Popov, A. Y., Sedletskii, A. M., Distribution of roots of Mittag-Leffler functions, Journal of Mathematical Sciences, 190(2) (2013), 209–409.
-
[35]
Ruzsa, I. Z., Székely, G. J., Convolution quotients of nonnegative functions, Monatshefte fÜr Mathematik, 95(3) (1983), 235–239.
-
[36]
Sato, K., Lévy processes and infinitely divisible distributions, Revised edition, Cambridge University Press, Cambridge, 2013.
-
[37]
Sato, K., Stochastic integrals with respect to Lévy processes and infinitely divisible distributions, Sugaku Expositions, 27 (2014), 19–42.
-
[38]
Székely, G. J., Half of a coin: negative probabilities, Wilmott Magazine (2005), 66–68.
-
[39]
van Haagen, A. J., Finite signed measures on function spaces, Pacific Journal of Mathematics, 95(2) (1981), 467–482.
-
[40]
van Haagen, A. J., Classifying infinitely divisible distributions by functional equations, Amsterdam: Mathematisch (1978).
-
[41]
Vellaisamy, P., Maheshwari, A., Fractional negative binomial and Polya processes. arXiv preprint arXiv:1306.2493, to appear in Probability and Mathematical Statistics.
-
[42]
Zhang, H., Liu, Y., Li, B., Notes on discrete compound Poisson model with applications to risk theory, Insurance: Mathematics and Economics, 59 (2014), 325–336.
-
[43]
Zhang, H., Li, B., Characterizations of discrete compound Poisson distributions, Communications in Statistics-Theory and Methods, 45(22) (2016), 6789–6802.
-
[44]
Zygmund, A., Trigonometric series, third edition, Vol I, II, Cambridge University Press, Cambridge, 2002.
-
[45]
Zhang, H., Wu, X., Notes on discrete compound Poisson point process and its concentration inequalities, arXiv preprint arXiv:1709.04159.
Monthly Content Usage
| Abstract Views | Full Text Views | PDF Downloads | |
|---|---|---|---|
| Aug 2020 | 6 | 1 | 2 |
| Sep 2020 | 13 | 0 | 0 |
| Oct 2020 | 6 | 0 | 0 |
| Nov 2020 | 5 | 0 | 0 |
| Dec 2020 | 0 | 0 | 0 |
| Jan 2021 | 7 | 0 | 0 |
| Feb 2021 | 0 | 0 | 0 |
Copyright Akadémiai Kiadó
AKJournals is the trademark of Akadémiai Kiadó's journal publishing business branch.
Copyright Akadémiai Kiadó AKJournals is the trademark of Akadémiai Kiadó's journal publishing business branch.
Powered by PubFactory