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

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.