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

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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 Published: 30 August 2013

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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), 2013. 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

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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

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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

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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

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@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}
}

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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  -

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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

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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

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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

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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  -

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