Power series with i.i.d. coefficients

In this work we consider a power series of the form X=∑j=0∞δjZj where 0<δ<10<δ<1 and {Zj}j≥0{Zj}j≥0 is an i.i.d. sequence of random variables. We show that XX is well-defined iff E[(log|Z0|)+]<∞E[(log|Z0|)+]<∞ and establish a number of properties of the distribution of XX, such as continuity and closure under convolution and weak convergence.