Electronic Communications in Probability

Absolute continuity of complex martingales and of solutions to complex smoothing equations

Ewa Damek and Sebastian Mentemeier

Full-text: Open access

Abstract

Let $X$ be a $\mathbb{C} $-valued random variable with the property that \[X \ \text{ has the same law as } \ \sum _{j\ge 1} T_j X_j\] where $X_j$ are i.i.d. copies of $X$, which are independent of the (given) $\mathbb{C} $-valued random variables $ (T_j)_{j\ge 1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins’ martingale with complex parameter.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 60, 12 pp.

Dates
Received: 6 April 2018
Accepted: 21 July 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1536718013

Digital Object Identifier
doi:10.1214/18-ECP155

Mathematical Reviews number (MathSciNet)
MR3863916

Zentralblatt MATH identifier
1401.60068

Subjects
Primary: 60G30: Continuity and singularity of induced measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60E10: Characteristic functions; other transforms 60G42: Martingales with discrete parameter

Keywords
absolute continuity branching process characteristic function complex smoothing equation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Damek, Ewa; Mentemeier, Sebastian. Absolute continuity of complex martingales and of solutions to complex smoothing equations. Electron. Commun. Probab. 23 (2018), paper no. 60, 12 pp. doi:10.1214/18-ECP155. https://projecteuclid.org/euclid.ecp/1536718013


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