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.
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