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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Using additive functionals to embed preassigned distributions in symmetric stable processes


Author: Itrel Monroe
Journal: Trans. Amer. Math. Soc. 163 (1972), 131-146
MSC: Primary 60J55
DOI: https://doi.org/10.1090/S0002-9947-1972-0298768-8
MathSciNet review: 0298768
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Abstract: Following Skorokhod, several authors in recent years have proposed methods to define a stopping time $ T$ for Brownian motion $ ({X_t},{\mathcal{F}_t})$ such that $ {X_T}$ will have some preassigned distribution. In this paper a method utilizing additive functionals is explored. It is applicable not only to Brownian motion but all symmetric stable processes of index $ \alpha > 1$. Using this method one is able to obtain any distribution having a finite $ \alpha - 1$ absolute moment. There is also a discussion of the problem of approximating symmetric stable processes with random walks.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0298768-8
Keywords: Symmetric stable processes, additive functionals, stopping times, martingales, preassigned distributions
Article copyright: © Copyright 1972 American Mathematical Society