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2018 Absolute continuity of complex martingales and of solutions to complex smoothing equations
Ewa Damek, Sebastian Mentemeier
Electron. Commun. Probab. 23(none): 1-12 (2018). DOI: 10.1214/18-ECP155

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.

Citation

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Ewa Damek. Sebastian Mentemeier. "Absolute continuity of complex martingales and of solutions to complex smoothing equations." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP155

Information

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

zbMATH: 1401.60068
MathSciNet: MR3863916
Digital Object Identifier: 10.1214/18-ECP155

Subjects:
Primary: 60G30, 60J80
Secondary: 60E10, 60G42

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