Fixed points of the smoothing transformation

Abstract

Let W1,..., WN be N nonnegative random variables and let {Mathematical expression} be the class of all probability measures on [0, ∞). Define a transformation T on {Mathematical expression} by letting Tμ be the distribution of W1X1+ ... + WN XN, where the Xi are independent random variables with distribution μ, which are independent of W1,..., WN as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for T to have a nontrivial fixed point of finite mean in the special cases that the Wi are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some γ>1, EWiγ