Symmetric fixed points of a smoothing transformation

Abstract

LetT= (T1,T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|>0< ∞ almost surely. We consider the following equation for distributions μ:W≅ ∑j=1∞TjWj, whereW,W1,W2,… have distribution μ andT,W1,W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions forTj≥ 0: essentially under the condition that E ∑j=1∞Tj2log+Tj2< ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.