On Compositions of Random Functions on a Finite Set

We establish the limiting distribution of the number Tn of random functions on a set of size n which must be composed before a constant function results. In more detail, let f1, f2, . . . be independent draws from the uniform distribution over all functions from {1, . . . , n} into itself. For t = 1, 2, . . . let gt := ft ◦ · · · ◦ f2 ◦ f1 denote the composition of the first t random maps, and let Tn be the smallest t such that gt is constant. Then Tn/n converges in distribution, with convergence of moment generating functions (and hence of all moments), to the infinite convolution of exponential distributions with rates