We study in a systematic form the contractive behavior of the map S of distributions to distributions S(F) = D∑iTiXi+C,(C,T=(T1,T2...)), Xi are independent r.v., L(Xi) = F. Further we show higher and exponential moments of the fixed point. Applications of this structure are given for (a) weighted branching processes, (b) the Hausdorff dimension of random Cantor sets and (c) the sorting algorithm Quicksort. © 1992.
Rösler, U. (1992). A fixed point theorem for distributions. Stochastic Processes and Their Applications, 42(2), 195–214. https://doi.org/10.1016/0304-4149(92)90035-O