The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sucient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach of Lyons, Peres and Pemantle (1995) to this theorem, which exploits a change of measure argument, is extended to martingales dened on Galton-Watson processes with a general type space through non-negative functions that are harmonic for the mean kernel. Many examples satisfy stochastic domination conditions on the ospring distributions that combine with the measure change argument to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem; a general treatment of this phenomenon is given. The application of the approach to branching processes in varying environments and random environments is indicated; the results also apply to the general (Crump-Mode-Jagers) branching process once suitable results on what are called optional lines are obtained. However, the main reason for developing the theory was to obtain martingale convergence results in branching random walk that did not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application. 1

This paper studies the behavior of RWRE on trees in the critical case left open in previous work. For trees of exponential growth, a random perturbation of the transition probabilities can change a transient random walk into a recurrent one. This is the opposite of what occurs on trees of sub-exponential growth. 1

Evans (1992) defines a notion of what it means for a set B to be polar for a process indexed by a tree,. The main result herein is that a tree picked from a Galton-Watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies that deterministic and nondeterministic trees behave identically in a variety of probability models. Mapping subsets of Euclidean space to trees and polar sets to capacity criteria, it also follows that certain random Cantor sets are capacity-equivalent to each other and to deterministic Cantor sets.

Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean p ≤ 1/2. How many of these Bernoullis one must look at in order to find a path of length n from the root which maximizes, up to a factor of 1 −ε, the sum of the Bernoullis along the path? In the case p = 1/2 (the critical value for nontriviality), it is shown to take Θ(ε −1 n) steps. In the case p < 1/2, the number of steps is shown to be at least n · exp(constε −1/2). This last result matches the known upper bound from [Algorithmica 22 (1998) 388–412] in a certain family of subcases.

Let S(v) be a function defined on the vertices v of the infinite binary tree. One algorithm to seek large positive values of S is the Metropolis-type Markov chain (Xn ) defined by P (Xn+1 = wjXn = v) = 1 3 e b(S(w)\GammaS(v)) 1 + e b(S(w)\GammaS(v)) for each neighbor w of v, where b is a parameter ("1=temperature") which the user can choose. We introduce and motivate study of this algorithm under a probability model for the objective function S, in which S is "tree-indexed simple random walk", that is the increments ¸(e) = S(w) \Gamma S(v) along parent-child edges e = (v; w) are independent and P (¸ = 1) = p; P (¸ = \Gamma1) = 1 \Gamma p. This algorithm has a "speed" r(p; b) = lim n n \Gamma1 ES(Xn ). We study the speed via a mixture of rigorous arguments, non-rigorous arguments and Monte Carlo simulations, and compare with a deterministic greedy algorithm which permits rigorous analysis. Formalizing the non-rigorous arguments presents a challenging problem. Mathematically, th...

We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite [26, 31]. In this paper, we confirm a conjecture of Aldous [3, 4] that E [Z] < ∞ while E [Z log Z] = ∞. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks. 1

This article examines a recent body of work on stochastic processes indexed by a tree. Emphasis is on the application of this new framework to existing probability models. Proofs are largely omitted, with references provided.

This paper considers a problem in extreme value theory from a computational complexity viewpoint. Suppose that {Snk: n ≥ 1, k ≤ K(n)} are random variables, with K(n) growing perhaps quite rapidly. Let Mn: = max k≤K(n) Snk. A prototypical classical extreme value theorem takes the form fn(Mn) → Z, where convergence is to a constant or a distribution. When K(n) grows rapidly with

Asymptotics for the survival probability in a killed branching random walk by

Asymptotics for the survival probability in a supercritical branching random walk by