TY - JOUR

T1 - A new family of mappings of infinitely divisible distributions related to the goldie–steutel–bondesson class

AU - Aoyama, Takahiro

AU - Lindner, Alexander

AU - Maejima, Makoto

PY - 2010/1/1

Y1 - 2010/1/1

N2 - Let (X(μ)t ≥ 0) be a Levy 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 ∫10 log 1/t dX(μ)t 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 ∫10 (log 1/t)1/α dX(μ)t 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.

AB - Let (X(μ)t ≥ 0) be a Levy 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 ∫10 log 1/t dX(μ)t 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 ∫10 (log 1/t)1/α dX(μ)t 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.

KW - Compound Poisson process

KW - Infinitely divisible distribution

KW - Limit of the ranges of the iterated mappings

KW - Stochastic integral mapping

KW - The Goldie-Steutel-Bondesson class

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U2 - 10.1214/EJP.v15-791

DO - 10.1214/EJP.v15-791

M3 - Article

AN - SCOPUS:77955456881

SN - 1083-6489

VL - 15

SP - 1119

EP - 1142

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

ER -