On the asymptotics of moments of linear random recurrences

We propose a new method of analyzing the asymptotics of moments of certain linear random recurrences which is based on the technique of iterative functions. By using the method, we show that the moments of the number of collisions and the absorption time in the Poisson–Dirichlet coalescent behave like the powers of the "log star" function which grows slower than any iteration of the logarithm, and thereby we prove a weak law of large numbers. Finally, we discuss merits and limitations of the method and give several examples related to beta coalescents, recursive algorithms, and random trees.