We show that the Harris-Sevast'yanov transformation for supercritical Galton-Watson processes with positive extinction probability q can be modified in such a way that the extinction probability of the new process takes any value between 0 and q. We give a probabilistic interpretation for the new process. This note is closely related to Athreya and Ney (1972), Chapter 1.12.

A random rooted labelled tree on n vertices has asymptotically the same shape as a branching-type process, in which each generation of a branching process with Poisson family sizes, parameter one, is supplemented by a single additional member added at random to one of the families in that generation. In this note we use this probabilistic representation to deduce the asymptotic distribution of the distance from the root to the nearest endertex other than itself.

A class of Markov chains is considered for which a certain property of the tail events makes bounded harmonic functions obtainable from bounded space-time harmonic functions. Applications to almost surely convergent Markov chains are given and, in particular, a representation of Martin-Doob-Hunt type is derived for all bounded harmonic functions of a finite mean supercritical branching process.

In this paper X(t) denotes Brownian motion on the line 0 ≤ t < ∞, E is a compact subset of (0, ∞) and F a compact subset of ( -∞, ∞). Then δ(E, F) is the supremum of the numbers c such that

In [4, 5] some lower and upper bounds for δ were found in terms of dim E and dim F, and it seemed possible that δ(E, F) could be determined entirely by these constants; this much is false, as the examples in our last paragraph will demonstrate. Here we shall show that δ(E, F) depends on a certain metric character η(E × F). However, η is not calculated relative to the Euclidean metric: the set F must be compressed to compensate for the oscillations of most paths X(t). Fortunately, η(E × F) can be calculated for a large enough class of sets E and F, by means of sequences of integers, to test any conjectures (and disprove most of them). In passing from η to δ we present a slight variation of Frostman's theory [3II].