Tempered Stable Distributions: Stochastic Models for Multiscale Processes
This brief is concerned with tempered stable distributions and their associated Levy processes. It is a good text for researchers interested in learning about tempered stable distributions.
A tempered stable distribution is one which takes a stable distribution and modifies its tails to make them lighter. The motivation for this class comes from the fact that infinite variance stable distributions appear to provide a good fit to data in a variety of situations, but the extremely heavy tails of these models are not realistic for most real world applications. The idea of using distributions that modify the tails of stable models to make them lighter seems to have originated in the influential paper of Mantegna and Stanley (1994). Since then, these distributions have been extended and generalized in a variety of ways. They have been applied to a wide variety of areas including mathematical finance, biostatistics,computer science, and physics.
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ˇ ˇ ˇ assume Borel function Borel set characteristic function cumulant define Definition denote density discuss distribution on Rd domain of attraction dominated convergence ETSp exists exponential extended Rosiński measure fact finite Borel measure finite measure Fix c e Gaussian given holds implies infinite variance stable infinitely divisible distributions integral jxj1 jxj2R.dx jxj─1 jZij Lemma LÚvy flight LÚvy measure lim sup limt!c Michael Grabchak 2016 models Monotone Convergence Theorem NRd0 open set p-tempered stable LÚvy parameter polar decomposition probability measure process with XI proper p-tempered O-stable Proposition 2.6 R.dx R(dx Radon measures random variables regularly varying relatively compact risk-neutral risk-neutral measure Rosiniski satisfies sequence short time behavior ſ║ SpringerBriefs in Mathematics stable LÚvy processes STLF tempered stable distributions vague convergence variance stable distribution weak convergence write