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

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.