  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 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.

Probability;Classical Analysis and ODEs