On finite truncation of infinite shot noise series representation of tempered stable laws

Junichi Imai, Reiichiro Kawai

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)


Tempered stable processes are widely used in various fields of application as alternatives with finite second moment and long-range Gaussian behaviors to stable processes. Infinite shot noise series representation is the only exact simulation method for the tempered stable process and has recently attracted attention for simulation use with ever improved computational speed. In this paper, we derive series representations for the tempered stable laws of increasing practical interest through the thinning, rejection, and inverse Lévy measure methods. We make a rigorous comparison among those representations, including the existing one due to Imai and Kawai [29] and Rosiski (2007) [3], in terms of the tail mass of Lévy measures which can be simulated under a common finite truncation scheme. The tail mass are derived in closed form for some representations thanks to various structural properties of the tempered stable laws. We prove that the representation via the inverse Lévy measure method achieves a much faster convergence in truncation to the infinite sum than all the other representations. Numerical results are presented to support our theoretical analysis.

Original languageEnglish
Pages (from-to)4411-4425
Number of pages15
JournalPhysica A: Statistical Mechanics and its Applications
Issue number23-24
Publication statusPublished - 2011 Nov 1


  • Infinitely divisible random vector
  • Inverse Lévy measure method
  • Rejection method
  • Sample path simulation
  • Shot noise method
  • Tempered stable process
  • Thinning method

ASJC Scopus subject areas

  • Statistics and Probability
  • Condensed Matter Physics


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