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 Electronic Communications in Probability > Vol. 15(2010) > Paper 21 open journal systems 


Compositions of mappings of infinitely divisible distributions with applications to finding the limits of some nested subclasses

Makoto Maejima, Keio University
Yohei Ueda, Keio University


Abstract
Let {Xt(μ),t≥ 0} be a Lévy process on Rd whose distribution at time 1 is μ, and let f be a nonrandom measurable function on (0, a), 0<a≤ ∞. Then we can define a mapping Φf(μ) by the law of ∫0af(t)dXt(μ), from D(Φf) which is the totality of μ∈ I(Rd) such that the stochastic integral ∫0af(t)dXt(μ) is definable, into a class of infinitely divisible distributions. For m∈N, let Φfm be the m times composition of Φf itself. Maejima and Sato (2009) proved that the limits ∩m=1∞Φfm(D(Φfm)) are the same for several known f's. Maejima and Nakahara (2009) introduced more general f's. In this paper, the limits ∩m=1∞Φfm(D(Φfm)) for such general f's are investigated by using the idea of compositions of suitable mappings of infinitely divisible distributions.


Full text: PDF

Pages: 227-239

Published on: June 28, 2010
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Electronic Communications in Probability. ISSN: 1083-589X