Random walks on trees and an inequality of means (Q1266771)

In a rooted labeled tree generated by a bi-infinite sequence \((d_i)\), each vertex belonging to generation \(i\) has \(d_i\) successors, and a random walk on such a tree is a random sequence of successive neighboring vertices starting from the root. A distribution on the set of bi-infinite sequences is interpreted as a stationary distribution for random walks on random directed rooted labeled trees. Then, a simple random walk on a tree generated by a bi-infinite sequence is compared with a simple random walk on an augmented Galton-Watson tree. Comparable simple random walks require the augmented Galton-Watson tree to be larger than the corresponding tree generated by a bi-infinite sequence, and this is due to an equality for random variables involving harmonic, geometric and arithmetic means.