### Abstract

The martingale limit law of the supercritical continuous time and state branching process either is compound Poisson or self-decomposable. This paper explores some general aspects of the latter case. A fundamental question for the latter case is whether the cumulant function of the martingale limit is a Thorin–Bernstein function. We make some progress by showing that it is special Bernstein if the cumulant function of the generating subordinator is special Bernstein. A specific parametric family of martingale limit cumulant functions is shown to be Thorin–Bernstein. Two complementary proofs of this fact are offered, one of which entirely avoids complex variable issues. The principal Lambert W-function is a boundary case of this family, thereby giving a new proof that it too is Thorin–Bernstein. Tail estimates of the distribution functions for this family are derived along with the right-hand tail and integral representations of their Lévy densities.

Original language | English |
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Journal | Journal of Theoretical Probability |

DOIs | |

Publication status | E-pub ahead of print - 26 Feb 2019 |

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**Self-Decomposable Laws from Continuous Branching Processes.** / Pakes, Anthony G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Self-Decomposable Laws from Continuous Branching Processes

AU - Pakes, Anthony G.

PY - 2019/2/26

Y1 - 2019/2/26

N2 - The martingale limit law of the supercritical continuous time and state branching process either is compound Poisson or self-decomposable. This paper explores some general aspects of the latter case. A fundamental question for the latter case is whether the cumulant function of the martingale limit is a Thorin–Bernstein function. We make some progress by showing that it is special Bernstein if the cumulant function of the generating subordinator is special Bernstein. A specific parametric family of martingale limit cumulant functions is shown to be Thorin–Bernstein. Two complementary proofs of this fact are offered, one of which entirely avoids complex variable issues. The principal Lambert W-function is a boundary case of this family, thereby giving a new proof that it too is Thorin–Bernstein. Tail estimates of the distribution functions for this family are derived along with the right-hand tail and integral representations of their Lévy densities.

AB - The martingale limit law of the supercritical continuous time and state branching process either is compound Poisson or self-decomposable. This paper explores some general aspects of the latter case. A fundamental question for the latter case is whether the cumulant function of the martingale limit is a Thorin–Bernstein function. We make some progress by showing that it is special Bernstein if the cumulant function of the generating subordinator is special Bernstein. A specific parametric family of martingale limit cumulant functions is shown to be Thorin–Bernstein. Two complementary proofs of this fact are offered, one of which entirely avoids complex variable issues. The principal Lambert W-function is a boundary case of this family, thereby giving a new proof that it too is Thorin–Bernstein. Tail estimates of the distribution functions for this family are derived along with the right-hand tail and integral representations of their Lévy densities.

KW - Continuous state branching

KW - Convolution equivalence

KW - Lambert W-function

KW - Self-decomposable laws

KW - Special and Thorin–Bernstein functions

KW - Stable laws

KW - Subordinators

UR - http://www.scopus.com/inward/record.url?scp=85062775758&partnerID=8YFLogxK

U2 - 10.1007/s10959-019-00886-0

DO - 10.1007/s10959-019-00886-0

M3 - Article

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

ER -