Abstract
In Chap. 2, we established a “log normal“ limit law yielding the power Poisson processes \(\mathcal {E}_{+}\) and \(\mathcal {E}_{-}\). In this chapter, we present five more limit laws that yield these power Poisson processes: three limit laws that are based on the linear renormalization schemes of Chap. 10, and two limit laws that are based on the power evolution schemes of Chap. 12.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
More specifically, the asymptotic equivalences of Sect. 13.4, \(\lambda \left( x\right) \approx \lambda _{+}\left( x\right) \) (\(x\rightarrow \infty \)) and \(\lambda \left( x\right) \approx \lambda _{-}\left( x\right) \) (\(x\rightarrow 0\)), imply that the intensity function \(\lambda \left( x\right) \) (\(x>0\)) is non-integrable over the positive half-line: \(\int _{0}^{\infty }\lambda \left( x\right) dx=\infty \). Hence, we cannot replace the intensity function \(\lambda \left( x\right) \) (\(x>0\)) with a probability density function f(x) (\(x>0\)). The reason why such a replacement worked out in the transition from Sects. 13.1 to 13.2 is that there the asymptotic equivalences were “flipped”—\(\lambda \left( x\right) \approx \lambda _{+}\left( x\right) \) (\(x\rightarrow 0\)) and \(\lambda \left( x\right) \approx \lambda _{-}\left( x\right) \) (\(x\rightarrow \infty \))—thus causing no integrability issue in Sect. 13.2.
References
N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation (Cambridge University Press, Cambridge, 1987)
I. Eliazar, Europhys. Lett. 119, 60007 (2017)
J.F.C. Kingman, Poisson Processes (Oxford University Press, Oxford, 1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Eliazar, I. (2020). Limit Laws. In: Power Laws. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-33235-8_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-33235-8_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-33234-1
Online ISBN: 978-3-030-33235-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)