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arXiv · 1804.02209

Absolute Continuity of Complex Martingales and of Solutions to Complex Smoothing Equations

Ewa Damek·Sebastian Mentemeier
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Abstract

Let $X$ be a $\mathbb{C}$-valued random variable with the property that $$X \ \text{ has the same law as }\ \sum_{j\ge1} T_j X_j$$ where $X_j$ are i.i.d.\ copies of $X$, which are independent of the (given) $\mathbb{C}$-valued random variables $ (T_j)_{j\ge1}$. 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.

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Ewa Damek, Sebastian Mentemeier. 2018-04-06. Absolute Continuity of Complex Martingales and of Solutions to Complex Smoothing Equations. https://arxiv.org/abs/1804.02209

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