On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies

Huiming Zhang; Jiao He; Hanlin Huang

doi:doi:10.11648/j.ajtas.20130205.11

| Peer-Reviewed

On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies

Huiming Zhang, Jiao He, Hanlin Huang

Published in American Journal of Theoretical and Applied Statistics (Volume 2, Issue 5)

Received: 3 August 2013 Accepted: Published: 30 August 2013

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Discrete compound Poisson processes (namely nonnegative integer-valued Lévy processes) have the property that more than one event occurs in a small enough time interval. These stochastic processes produce the discrete compound Poisson distributions. In this article, we introduce ten approaches to prove the probability mass function of discrete compound Poisson distributions, and we obtain seven approaches to prove the probability mass function of Poisson distributions. Finally, we discuss the connection between additive functions in probabilistic number theory and discrete compound Poisson distributions and give a numerical example. Stuttering Poisson distributions (a special case of discrete compound Poisson distributions) are applied to numerical solution of optimal (s, S) inventory policies by using continuous approximation method.

Published in	American Journal of Theoretical and Applied Statistics (Volume 2, Issue 5)
DOI	10.11648/j.ajtas.20130205.11
Page(s)	110-121
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Probability Mass Function, Nonnegative Integer-Valued Lévy Processes, Probabilistic Number Theory, Discrete Compound Poisson Distribution, (S, S) Inventory Policies

References

[1]	Adelson R M. Compound poisson distributions. Operational Research.1966,17(1):73-75.
[2]	Barndorff-Nielsen O E, Pollard D G, Shephard N. Integer-valued Lévy processes and low latency financial econometrics. Quantitative Finance. 2012;12(4):587-605.
[3]	Bekelis D. Convolutions of the Poisson laws in number theory, In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania, 23-27 September 1996 (Vol. 4, p. 283). Netherlands:VSP BV,1997.
[4]	Bell E T. Exponential polynomials. The Annals of Mathematics. 1934,35(2): 258-277.
[5]	Bertoin, J. Lévy processes. London: Cambridge university press,1998.
[6]	De Finetti, B. Sulla possibilita di valori eccezionali per una legge ad incrementi aleatori. Atti della Reale Accademina Nazionale dei Lincei (Ser VI). 1929,10:325-329.
[7]	De Finetti, B. Le funzioni caratteristiche di legge istantanea, Rendicontidella R. Accademia Nazionale dei Lincei (Ser VI). 1930,12:278-282.
[8]	De Koninck J M, Galambos J. The intermediate prime divisors of integers. Proceedings of the American Mathematical Society. 1987,101(2):213-216.
[9]	Erdös P, Wintner A. Additive arithmetical functions and statistical independence. American Journal of Mathematics. 1939,61(3):713-721.
[10]	Faa di Bruno, CF. Sullo sviluppo delle funzioni. Annali di Scienze Matematiche e Fisiche. 1855,6:479-480.
[11]	Gallagher PX. On the distribution of primes in short intervals. Mathematika. 1976,23(1): 4-9.
[12]	Galliher H P, Morse P M, Simond M. Dynamics of two classes of continuous-review inventory systems. Operations Research. 1959,7(3):362-384.
[13]	Granville A. Prime divisors are Poisson distributed. International Journal of Number Theory. 2007,3(01):1-18.
[14]	Haight FA. Handbook of the Poisson distribution. Los Angeles: Wiley;1967.
[15]	Halász G. On the distribution of additive and the mean-values of multiplicative arithmetic functions. Studia. Sci. Math. Hungarica. 1971;6:211–233.
[16]	Hofmann M. Über zusammengesetzte Poisson-Prozesse und ihre Anwendungen in der Unfallversicherung. Diss. Math. ETH Zürich, Nr. 2511, 0000. Ref.: Saxer, W.; Korref.: Nolfi, P., 1955.
[17]	Jánossy L, Rényi A, Aczél J. On composed Poisson distributions, I. Acta Mathematica Hungarica. 1950,1(2):209-224.
[18]	Johnson N L, Kemp A W, Kotz S. Univariate discrete distributions. New Jersey: Wiley ; 2005.
[19]	Kemp C D, Kemp A W. Some properties of the ‘Hermite’ distribution. Biometrika. 1965,52(3-4):381-394.
[20]	Khintchine AY. A new derivation of a formula by P. Lévy. Bulletin of the Moscow State University. 1937,1:1-5. [Russian]
[21]	Khintchine AY. Mathematical methods of queuing theory. Proceedings of the Steklov Institute. VA Steklov, 1955,49(0):3-122. [Russian]
[22]	Kolmogorov A N. Sulla forma generale di un processo stocastico omogeneo. Atti Accad. Naz. Lincei. 1932,15: 805-808.
[23]	Lévy P. Sur les intégrales dont les éléments sont des variables aléatoires indépendantes. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze. 1934,3(3-4):337-366.
[24]	Lüders R. Die statistik der seltenen ereignisse. Biometrika. 1934,26(1/2):108-128.
[25]	Poisson SD. Recherches sur la probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilités. Paris: Bachelier,1837.
[26]	Porteus EL. Foundations of stochastic inventory theory. Stanford: Stanford University Press,2002.
[27]	Privault N. Stochastic Analysis in Discrete and Continuous Settings With Normal Martingales. Berlin: Springer, 2009.
[28]	Šiaulys J. The von Mises theorem in number theory. New Trends in Probability and Statistics. 1992,2:293-310.
[29]	Tenenbaum G. Introduction to analytic and probabilistic number theory. London: Cambridge university press,1995.
[30]	Veinott A F, Wagner H M. Computing optimal (s, S) inventory policies. Management Science. 1965,11(5):525-552.
[31]	Voinov V G, Nikulin M. On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications. Kybernetika. 1994,30(3):343-358.
[32]	Wimmer G, Altmann G. The multiple poisson distribution, its characteristics and a variety of forms[J]. Biometrical journal. 1996, 38(8): 995-1011.
[33]	Whittaker J M. The shot effect for showers. Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, 1937,33(04):451-458.
[34]	Zhang H, Chu L, Diao Y. Some Properties of the Generalized Stuttering Poisson Distribution and Its Applications. Studies in Mathematical Sciences. 2012,5(1) :11-26.

Cite This Article

Plain Text BibTeX RIS

APA Style

Huiming Zhang, Jiao He, Hanlin Huang. (2013). On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics, 2(5), 110-121. https://doi.org/10.11648/j.ajtas.20130205.11

Copy | Download

ACS Style

Huiming Zhang; Jiao He; Hanlin Huang. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. Am. J. Theor. Appl. Stat. 2013, 2(5), 110-121. doi: 10.11648/j.ajtas.20130205.11

Copy | Download

AMA Style

Huiming Zhang, Jiao He, Hanlin Huang. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. Am J Theor Appl Stat. 2013;2(5):110-121. doi: 10.11648/j.ajtas.20130205.11

Copy | Download

@article{10.11648/j.ajtas.20130205.11,
  author = {Huiming Zhang and Jiao He and Hanlin Huang},
  title = {On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {2},
  number = {5},
  pages = {110-121},
  doi = {10.11648/j.ajtas.20130205.11},
  url = {https://doi.org/10.11648/j.ajtas.20130205.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20130205.11},
  abstract = {Discrete compound Poisson processes (namely nonnegative integer-valued Lévy processes) have the property that more than one event occurs in a small enough time interval. These stochastic processes produce the discrete compound Poisson distributions. In this article, we introduce ten approaches to prove the probability mass function of discrete compound Poisson distributions, and we obtain seven approaches to prove the probability mass function of Poisson distributions. Finally, we discuss the connection between additive functions in probabilistic number theory and discrete compound Poisson distributions and give a numerical example. Stuttering Poisson distributions (a special case of discrete compound Poisson distributions) are applied to numerical solution of optimal (s, S) inventory policies by using continuous approximation method.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies
AU  - Huiming Zhang
AU  - Jiao He
AU  - Hanlin Huang
Y1  - 2013/08/30
PY  - 2013
N1  - https://doi.org/10.11648/j.ajtas.20130205.11
DO  - 10.11648/j.ajtas.20130205.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 110
EP  - 121
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20130205.11
AB  - Discrete compound Poisson processes (namely nonnegative integer-valued Lévy processes) have the property that more than one event occurs in a small enough time interval. These stochastic processes produce the discrete compound Poisson distributions. In this article, we introduce ten approaches to prove the probability mass function of discrete compound Poisson distributions, and we obtain seven approaches to prove the probability mass function of Poisson distributions. Finally, we discuss the connection between additive functions in probabilistic number theory and discrete compound Poisson distributions and give a numerical example. Stuttering Poisson distributions (a special case of discrete compound Poisson distributions) are applied to numerical solution of optimal (s, S) inventory policies by using continuous approximation method.
VL  - 2
IS  - 5
ER  -

Copy | Download

Author Information

Huiming Zhang

Dept. of Mathematics and Statistics, Central China Normal University, Wuhan, China
Jiao He

Dept. of Mathematics and Statistics, Central China Normal University, Wuhan, China
Hanlin Huang

Dept. of Mathematics and Statistics, Central China Normal University, Wuhan, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Huiming Zhang, Jiao He, Hanlin Huang. (2013). On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics, 2(5), 110-121. https://doi.org/10.11648/j.ajtas.20130205.11

Copy | Download

ACS Style

Huiming Zhang; Jiao He; Hanlin Huang. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. Am. J. Theor. Appl. Stat. 2013, 2(5), 110-121. doi: 10.11648/j.ajtas.20130205.11

Copy | Download

AMA Style

Huiming Zhang, Jiao He, Hanlin Huang. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. Am J Theor Appl Stat. 2013;2(5):110-121. doi: 10.11648/j.ajtas.20130205.11

Copy | Download

@article{10.11648/j.ajtas.20130205.11,
  author = {Huiming Zhang and Jiao He and Hanlin Huang},
  title = {On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {2},
  number = {5},
  pages = {110-121},
  doi = {10.11648/j.ajtas.20130205.11},
  url = {https://doi.org/10.11648/j.ajtas.20130205.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20130205.11},
  abstract = {Discrete compound Poisson processes (namely nonnegative integer-valued Lévy processes) have the property that more than one event occurs in a small enough time interval. These stochastic processes produce the discrete compound Poisson distributions. In this article, we introduce ten approaches to prove the probability mass function of discrete compound Poisson distributions, and we obtain seven approaches to prove the probability mass function of Poisson distributions. Finally, we discuss the connection between additive functions in probabilistic number theory and discrete compound Poisson distributions and give a numerical example. Stuttering Poisson distributions (a special case of discrete compound Poisson distributions) are applied to numerical solution of optimal (s, S) inventory policies by using continuous approximation method.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies
AU  - Huiming Zhang
AU  - Jiao He
AU  - Hanlin Huang
Y1  - 2013/08/30
PY  - 2013
N1  - https://doi.org/10.11648/j.ajtas.20130205.11
DO  - 10.11648/j.ajtas.20130205.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 110
EP  - 121
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20130205.11
AB  - Discrete compound Poisson processes (namely nonnegative integer-valued Lévy processes) have the property that more than one event occurs in a small enough time interval. These stochastic processes produce the discrete compound Poisson distributions. In this article, we introduce ten approaches to prove the probability mass function of discrete compound Poisson distributions, and we obtain seven approaches to prove the probability mass function of Poisson distributions. Finally, we discuss the connection between additive functions in probabilistic number theory and discrete compound Poisson distributions and give a numerical example. Stuttering Poisson distributions (a special case of discrete compound Poisson distributions) are applied to numerical solution of optimal (s, S) inventory policies by using continuous approximation method.
VL  - 2
IS  - 5
ER  -

Copy | Download