In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write

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.

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

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

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