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A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class
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Takahiro Aoyama, Tokyo University of Science
Alexander Lindner, Technische Universitat Braunschweig
Makoto Maejima, Keio University
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Abstract
Let {Xt(μ),t≥ 0} be a Lévy process on Rd whose distribution at time 1 is a
d-dimensional infinitely distribution μ.
It is known that the set of all infinitely divisible
distributions on Rd, each of which is represented by the law of a stochastic
integral
∫01log(1/t)dXt(μ)
for some infinitely divisible distribution on Rd, coincides with
the Goldie-Steutel-Bondesson class, which, in one dimension, is the smallest class that
contains all mixtures of exponential distributions and is closed under convolution and
weak convergence. The purpose of this paper is to study the class of infinitely divisible
distributions which are represented as the law of
∫01(log(1/t))1/αdXt(μ) for general
α> 0. These stochastic integrals define a new family of mappings of infinitely divisible
distributions. We first study properties of these mappings and their ranges. Then
we characterize some subclasses of the range by stochastic integrals with respect to
some compound Poisson processes. Finally, we investigate the limit of the ranges of
the iterated mappings.
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Full text: PDF
Pages: 1119-1142
Published on: July 7, 2010
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Bibliography
-
T. Aoyama.
Nested subclasses of the class of type G selfdecomposable distributions on Rd.
Probab. Math. Statist. 29 (2009), 135-154.
MR2553004
-
T. Aoyama, M. Maejima, and J. Rosiński.
A subclass of type G selfdecomposable distributions on Rd.
J. Theor. Probab. 21 (2008), 14-34.
MR2384471
-
O.E. Barndorff-Nielsen, M. Maejima, and K. Sato.
Some classes of multivariate infinitely divisible distributions
admitting stochastic integral representations.
Bernoulli 12 (2006), 1-33.
MR2202318
-
L. Bondesson.
Class of infinitely divisible distributions and densities.
Z. Wahrsch. Verw. Gebiete 57 (1981),39-71; Correction and addendum 59 (1982), 277.
MR623454
-
W. Feller.
An introduction to probability theory and its applications, Vol. II, 2nd ed.
John Wiley & Sons (1981).
MR0210154
-
L.F. James, B. Roynette and M. Yor.
Generalized Gamma convolutions, Dirichlet means, Thorin measures, with explicit examples.
Probab. Surv. 5 (2008), 346-415.
MR2476736
-
M. Maejima.
Subclasses of Goldie-Steutel-Bondesson class of infinitely
divisible distributions on Rd by Υ-mapping.
ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 55-66.
MR2324748
-
M. Maejima and G. Nakahara.
A note on new classes of infinitely divisible distributions on
Rd.
Electron. Commun. Probab. 14 (2009), 358-371.
MR2535084
-
M. Maejima and K. Sato.
The limits of nested subclasses of several classes of infinitely
divisible distributions are identical with the closure of the class of stable
distributions.
Probab. Theory Relat. Fields 145 (2009), 119-142.
MR2520123
-
K. Sato.
Class L of multivariate distributions and its subclasses.
J. Multivariate Anal. 10 (1980), 207-232.
MR575925
-
K. Sato.
Lévy Processes and Infinitely Divisible Distributions.
Cambridge University Press, Cambridge (1999).
MR1739520
-
K. Sato.
Stochastic integrals in additive processes and application to
semi-Lévy processes.
Osaka J. Math. 41 (2004), 211-236.
MR2040073
-
K. Sato.
Additive processes and stochastic integrals.
Illinois J. Math. 50 (2006). 825-851.
MR2247848
-
K. Sato.
Two families of improper stochastic integrals with respect to Lévy
processes.
ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 47-87.
MR2235174
-
K. Sato.
Transformations of infinitely divisible distributions via improper stochastic integrals.
ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 67-110.
MR2349803
-
S.J. Wolfe.
On a continuous analogue of the stochastic difference equation
Xn=ρ Xn-1+Bn.
Stochastic Process. Appl. 12 (1982), 301-312.
MR656279
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Electronic Journal of Probability. ISSN: 1083-6489 |
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