Fixed points of a generalized smoothing transformation and applications to the branching random walk

Let {A(i) : i greater than or equal to 1} be a sequence of non-negative random variables and let M be the class of all probability measures on [0, infinity]. Define a transformation T on M by letting T mu be the distribution of Sigma(i=1)(infinity) A(i)Z(i), where the Z(i) are independent random variables with distribution mu, which are also independent of {A(i)}. Under first moment assumptions imposed on {A(i)}, we determine exactly when T has a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + epsilon, epsilon > 0, we find all the fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (by T) with an appropriate initial distribution; convergence to the trivial fixed points Sg and delta(infinity) is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.