Tail probability and singularity of Laplace-Stieltjes transform of a Pareto type random variable

Kenji Nakagawa

Applications of Mathematics (2015)

  • Volume: 60, Issue: 2, page 157-184
  • ISSN: 0862-7940

Abstract

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We give a sufficient condition for a non-negative random variable X to be of Pareto type by investigating the Laplace-Stieltjes transform of the cumulative distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. For the proof of our theorems, we apply Graham-Vaaler’s complex Tauberian theorem. As an application of our theorems, we consider the asymptotic decay of the stationary distribution of an M/G/1 type Markov chain.

How to cite

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Nakagawa, Kenji. "Tail probability and singularity of Laplace-Stieltjes transform of a Pareto type random variable." Applications of Mathematics 60.2 (2015): 157-184. <http://eudml.org/doc/269874>.

@article{Nakagawa2015,
abstract = {We give a sufficient condition for a non-negative random variable $X$ to be of Pareto type by investigating the Laplace-Stieltjes transform of the cumulative distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. For the proof of our theorems, we apply Graham-Vaaler’s complex Tauberian theorem. As an application of our theorems, we consider the asymptotic decay of the stationary distribution of an M/G/1 type Markov chain.},
author = {Nakagawa, Kenji},
journal = {Applications of Mathematics},
keywords = {tail probability; Pareto type; Laplace-Stieltjes transform; Tauberian theorem; tail probability; Pareto type; Laplace-Stieltjes transform; Tauberian theorem},
language = {eng},
number = {2},
pages = {157-184},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Tail probability and singularity of Laplace-Stieltjes transform of a Pareto type random variable},
url = {http://eudml.org/doc/269874},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Nakagawa, Kenji
TI - Tail probability and singularity of Laplace-Stieltjes transform of a Pareto type random variable
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 2
SP - 157
EP - 184
AB - We give a sufficient condition for a non-negative random variable $X$ to be of Pareto type by investigating the Laplace-Stieltjes transform of the cumulative distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. For the proof of our theorems, we apply Graham-Vaaler’s complex Tauberian theorem. As an application of our theorems, we consider the asymptotic decay of the stationary distribution of an M/G/1 type Markov chain.
LA - eng
KW - tail probability; Pareto type; Laplace-Stieltjes transform; Tauberian theorem; tail probability; Pareto type; Laplace-Stieltjes transform; Tauberian theorem
UR - http://eudml.org/doc/269874
ER -

References

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  8. Moriguchi, K., al., et, A Table of Mathematical Formulas II, Iwanami Shoten (1957), Japanese. (1957) 
  9. Nakagawa, K., 10.1081/STM-120028389, Stoch. Models 20 (2004), 31-42. (2004) Zbl1035.60011MR2036294DOI10.1081/STM-120028389
  10. Nakagawa, K., 10.1080/00036810410001724436, Appl. Anal. 84 (2005), 499-522. (2005) Zbl1085.44002MR2151276DOI10.1080/00036810410001724436
  11. Nakagawa, K., 10.1109/TIT.2007.903114, IEEE Trans. Inf. Theory 53 (2007), 3239-3249. (2007) MR2417689DOI10.1109/TIT.2007.903114
  12. Rudin, W., Real and Complex Analysis, McGraw-Hill, New York (1987). (1987) Zbl0925.00005MR0924157
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