Conceptual Proofs of $L$ Log $L$ Criteria for Mean Behavior of Branching Processes

Abstract

The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution. §1. Introduction.Consider a Galton-Watson branching process with each particle having probability p k of generating k children. Let L stand for a random variable with this offspring distribution. Let m := k kp k be the mean number of children per particle and let Z n be the number of particles in the n th generation. The most basic and well-known fact about branching processes is that the extinction probability q := lim P[Z n = 0] is equal to 1 if and only if m ≤ 1 and p 1 < 1. It is also not hard to establish that in the case m > 1, 1 n log Z n → log m almost surely on nonextinction, while in the case m ≤ 1, 1 n log P[Z n > 0] → log m.Finer questions may be asked:• In the case m > 1, when does the mean E[Z n ] = m n give the right growth rate up to a random factor?• In the case m < 1, when does the first moment estimate P[Z n > 0] ≤ E[Z n ] = m n give the right decay rate up to a random factor?• In the case m = 1, what is the decay rate of P[Z n > 0]?These questions are answered by the following three classical theorems.1991 Mathematics Subject Classification. Primary 60J80.

A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales

“…The reader who is already familiar with the Lyons et al [41,43,44] papers will recall that they used two separate underlying spaces of marked trees with and without the spines, then marginalized out the spine when wanting to deal only with the branching particles as a whole. Instead, we are going to use the single underlying spaceT , but define four filtrations of it that will encapsulate different knowledge.…”

“…The phenomena of sizebiasing along the spine is a common feature of such measure changes when random offspring distributions are present. Although Chauvin and Rouault's work on the measure change continued in a paper co-authored with Wakolbinger [10], where the new measure is interpreted as the result of building a conditioned tree using the concepts of Palm measures, it wasn't until the so-called 'conceptual proofs' of Lyons, Kurtz, Peres and Pemantle published around 1995 ( [44,43,41]) that the spine approach really began to crystalize. These papers laid out a formal basis for spines using a series of new measures on two underlying spaces of sample trees with and without distinguished lines of descent (spines).…”

“…In this article 1 , we present a modified formalization of the spine approach that attempts to improve on the schemes originally laid out by Lyons et al [44,43,41] and later for BBM by Kyprianou [40]. Although the set-up costs of our spine formalization are quite large, at least in terms of definitions and notation, the underlying ideas are all extremely simple and intuitive.…”

“…The reader who is familiar with the work of Lyons et al [44] or Kyprianou [40] will notice significant similarities as well as differences in our approach. In the first instance our modifications correct our perceived weakness in the Lyons et al scheme where one of the measures they defined had a time-dependent mass and could not be normalized to be a probability measure in a natural way, hence lacked a clear interpretation in terms of any direct process construction; an immediate consequence of this improvement is that here all measure changes are carried out by martingales and we regain a clear intuitive construction.…”

“…We find the same conditions are also necessary for L p -boundedness of the martingale when p > 1 by just considering the contributions along the spine at times of fission and observing when these are unbounded. Otherwise, to determine whether the martingale is merely L 1 -convergent or has an almost-surely zero limit, we determine whether the martingale is almostsurely bounded or not under its own change of measure -this was Kyprianou's [40] approach and relies on a measure-theoretic result that has become standard in the spine methodology since the important work of Lyons et al [44,43,41].…”