Max-analogues of N-infinite Divisibility and N-stability

Satheesh Sreedharan; Sandhya E.

doi:doi:10.11648/j.ijsd.20150102.11

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Max-analogues of N-infinite Divisibility and N-stability

Satheesh Sreedharan, Sandhya E.

Published in International Journal of Statistical Distributions and Applications (Volume 1, Issue 2)

Received: 27 September 2015 Accepted: 14 November 2015 Published: 22 November 2015

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Abstract

Here we discuss the max-analogues of random infinite divisibility and random stability developed by Gnedenko and Korolev [5]. We give a necessary and sufficient condition for the weak convergence to a random max-infinitely divisible law from that to a max-infinitely divisible law. Introducing random max-stable laws we show that they are indeed invariant under random maximum. We then discuss their domain of max-attraction.

Published in	International Journal of Statistical Distributions and Applications (Volume 1, Issue 2)
DOI	10.11648/j.ijsd.20150102.11
Page(s)	33-36
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

Max-infinite Divisibility, Max-stability, Domain of Max-attraction, Extremal Processes

References

[1]	Klebanov, L. B; Maniya, G. M. and Melamed, I. A (1984). A problem of Zolotarev and analogues of infinitely divisible and stable distributions in the scheme of summing a random number of random variables, Theory of Probability and Applications, 29, 791 – 794.
[2]	Sandhya, E. (1991). Geometric Infinite Divisibility and Applications, Ph.D. Thesis (unpublished), University of Kerala, January 1991.
[3]	Sandhya, E. and Pillai, R. N. (1999). On geometric infinite divisibility, Journal of Kerala Statistical Association, 10, 1-7.
[4]	Mohan, N. R.; Vasudeva, R. and Hebbar, H. V. (1993). On geometrically infinitely divisible laws and geometric domains of attraction, Sankhya-A, 55, 171-179.
[5]	Gnedenko, B. V., Korolev, V. Y. (1996). Random summation: limit theorems and applications. CRC Press. Section 4.6, pp.137-152.
[6]	Klebanov, L. B. and Rachev, S. T. (1996). Sums of a random number of random variables and their approximations with ν-accompanying infinitely divisible laws, Serdica Math. Journal, 22, 471-496.
[7]	Sandhya, E. (1996). On a generalization of geometric infinite divisibility, Proc. 8th Kerala Science Congress, January-1996, 355-357.
[8]	Bunge, J. (1996). Composition semi groups and random stability, Annals of Probability, 24, 476-1489.
[9]	Satheesh, S; Sandhya, E. and Lovely T Abraham (2010). Limit distributions of random sums of Z⁺-valued random variables, Communications in Statistics—Theory and Methods, 39, 1979-1984.
[10]	Satheesh, S (2004). Another look at random infinite divisibility, Statistical Methods, 6(2), 123-144.
[11]	Balkema, A. A., Resnick, S. I. (1977). Max-infinite divisibility. Journal of Applied Probability, 309-319.
[12]	Rachev, S. T., Resnick, S. (1991). Max-geometric infinite divisibility and stability. Communications in Statistics. Stochastic Models, 7(2), 191-218.
[13]	Mohan, N. R. (1998). On geometrically max infinitely divisible laws. Journal of Indian Statistical Association, 36(1), 1-12.
[14]	Gnedenko, B. V. (1983). On limit theorems for a random number of random variables. In Probability Theory and Mathematical Statistics (pp. 167-176). Springer Berlin Heidelberg.
[15]	Satheesh, S., Sandhya, E., Rajasekharan, K. E. (2008). A generalization and extension of an autoregressive model. Statistics & Probability Letters, 78(12), 1369-1374.
[16]	Barakat, H. M., Ghitany, M. E., Al-Hussaini, E. K. (2009). Asymptotic distributions of order statistics and record values under the Marshall–Olkin parametrization operation. Communications in Statistics—Theory and Methods, 38(13), 2267-2273.

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Satheesh Sreedharan, Sandhya E. (2015). Max-analogues of N-infinite Divisibility and N-stability. International Journal of Statistical Distributions and Applications, 1(2), 33-36. https://doi.org/10.11648/j.ijsd.20150102.11

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ACS Style

Satheesh Sreedharan; Sandhya E. Max-analogues of N-infinite Divisibility and N-stability. Int. J. Stat. Distrib. Appl. 2015, 1(2), 33-36. doi: 10.11648/j.ijsd.20150102.11

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AMA Style

Satheesh Sreedharan, Sandhya E. Max-analogues of N-infinite Divisibility and N-stability. Int J Stat Distrib Appl. 2015;1(2):33-36. doi: 10.11648/j.ijsd.20150102.11

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@article{10.11648/j.ijsd.20150102.11,
  author = {Satheesh Sreedharan and Sandhya E.},
  title = {Max-analogues of N-infinite Divisibility and N-stability},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {1},
  number = {2},
  pages = {33-36},
  doi = {10.11648/j.ijsd.20150102.11},
  url = {https://doi.org/10.11648/j.ijsd.20150102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20150102.11},
  abstract = {Here we discuss the max-analogues of random infinite divisibility and random stability developed by Gnedenko and Korolev [5]. We give a necessary and sufficient condition for the weak convergence to a random max-infinitely divisible law from that to a max-infinitely divisible law. Introducing random max-stable laws we show that they are indeed invariant under random maximum. We then discuss their domain of max-attraction.},
 year = {2015}
}

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TY  - JOUR
T1  - Max-analogues of N-infinite Divisibility and N-stability
AU  - Satheesh Sreedharan
AU  - Sandhya E.
Y1  - 2015/11/22
PY  - 2015
N1  - https://doi.org/10.11648/j.ijsd.20150102.11
DO  - 10.11648/j.ijsd.20150102.11
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 33
EP  - 36
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20150102.11
AB  - Here we discuss the max-analogues of random infinite divisibility and random stability developed by Gnedenko and Korolev [5]. We give a necessary and sufficient condition for the weak convergence to a random max-infinitely divisible law from that to a max-infinitely divisible law. Introducing random max-stable laws we show that they are indeed invariant under random maximum. We then discuss their domain of max-attraction.
VL  - 1
IS  - 2
ER  -

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Author Information

Satheesh Sreedharan

Department of Applied Sciences, Vidya Academy of Science and Technology, Thalakkottukara, Thrissur, India
Sandhya E.

Department of Statistics, Prajyoti Niketan College, Pudukad, Thrissur, India

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APA Style

Satheesh Sreedharan, Sandhya E. (2015). Max-analogues of N-infinite Divisibility and N-stability. International Journal of Statistical Distributions and Applications, 1(2), 33-36. https://doi.org/10.11648/j.ijsd.20150102.11

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ACS Style

Satheesh Sreedharan; Sandhya E. Max-analogues of N-infinite Divisibility and N-stability. Int. J. Stat. Distrib. Appl. 2015, 1(2), 33-36. doi: 10.11648/j.ijsd.20150102.11

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AMA Style

Satheesh Sreedharan, Sandhya E. Max-analogues of N-infinite Divisibility and N-stability. Int J Stat Distrib Appl. 2015;1(2):33-36. doi: 10.11648/j.ijsd.20150102.11

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@article{10.11648/j.ijsd.20150102.11,
  author = {Satheesh Sreedharan and Sandhya E.},
  title = {Max-analogues of N-infinite Divisibility and N-stability},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {1},
  number = {2},
  pages = {33-36},
  doi = {10.11648/j.ijsd.20150102.11},
  url = {https://doi.org/10.11648/j.ijsd.20150102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20150102.11},
  abstract = {Here we discuss the max-analogues of random infinite divisibility and random stability developed by Gnedenko and Korolev [5]. We give a necessary and sufficient condition for the weak convergence to a random max-infinitely divisible law from that to a max-infinitely divisible law. Introducing random max-stable laws we show that they are indeed invariant under random maximum. We then discuss their domain of max-attraction.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Max-analogues of N-infinite Divisibility and N-stability
AU  - Satheesh Sreedharan
AU  - Sandhya E.
Y1  - 2015/11/22
PY  - 2015
N1  - https://doi.org/10.11648/j.ijsd.20150102.11
DO  - 10.11648/j.ijsd.20150102.11
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 33
EP  - 36
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20150102.11
AB  - Here we discuss the max-analogues of random infinite divisibility and random stability developed by Gnedenko and Korolev [5]. We give a necessary and sufficient condition for the weak convergence to a random max-infinitely divisible law from that to a max-infinitely divisible law. Introducing random max-stable laws we show that they are indeed invariant under random maximum. We then discuss their domain of max-attraction.
VL  - 1
IS  - 2
ER  -

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