Stochastic fixed-point equation and local dependence measure

We study solutions to the stochastic fixed point equation $X\stackrel{d}{=}AX+B$ where the coefficients $A$ and $B$ are nonnegative random variables. We introduce the “local dependence measure” (LDM) and its Legendre-type transform to analyze the left tail behavior of the distribution of $X$. We discuss the relationship of LDM with earlier results on the stochastic fixed point equation and we apply LDM to prove a theorem on a Fleming-Viot-type process.