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On the asymptotic behaviour of increasing self-similar Markov processes


 
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1. Title Title of document On the asymptotic behaviour of increasing self-similar Markov processes
 
2. Creator Author's name, affiliation, country María Emilia Caballero; Instituto de Matemeticas UNAM Mexico
 
2. Creator Author's name, affiliation, country Víctor Manuel Rivero; Centro de Investigación en Matemáticas, Guanajuato
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) self-similar Markov processes
 
3. Subject Subject classification 60G18; 60G17; 60B10
 
4. Description Abstract It has been proved by Bertoin and Caballero cite{BC2002} that a $1/\alpha$-increasing self-similar Markov process $X$ is such that $t^{-1/\alpha}X(t)$ converges weakly, as $t\to\infty,$ to a degenerate random variable whenever the subordinator associated to it via Lamperti's transformation has infinite mean. Here we prove that $\log(X(t)/t^{1/\alpha})/\log(t)$ converges in law to a non-degenerate random variable if and only if the associated subordinator has Laplace exponent that varies regularly at $0.$ Moreover, we show that $\liminf_{t\to\infty}\log(X(t))/\log(t)=1/\alpha,$ a.s. and provide an integral test for the upper functions of $\{\log(X(t)), t\geq 0\}.$ Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained. In particular, these allow us to establish estimates for the left tail of some exponential functionals of subordinators. Finally, some of the implications of these results in the theory of self-similar fragmentations are discussed.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) CONCyTEG (Council of Science and Technology of the state of Guanajuato, Mexico) and partially by the project PAPIITT-IN120605, UNAM
 
7. Date (YYYY-MM-DD) 2009-04-19
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/637
 
10. Identifier Digital Object Identifier 10.1214/EJP.v14-637
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 14
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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