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

TL;DR

This paper establishes a simple criterion for the absolute continuity of complex-valued solutions to certain smoothing equations, including complex martingales, based on moment conditions and known existence criteria.

Contribution

It introduces a new, straightforward criterion for absolute continuity of solutions to complex smoothing equations, extending to complex martingales with minimal additional assumptions.

Findings

Criterion applies to Biggins' complex martingale.

Absolute continuity depends on finiteness of first and second moments of nonzero weights.

The approach simplifies previous conditions for absolute continuity.

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.