Robin Pemantle

A survey of random processes with reinforcement

The models surveyed include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.

Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure

We consider simple random walk on the family tree T of a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary of T. Concretely, this implies that an exponentially small fraction of the nth level of T carries most of the harmonic measure. First-order asymptotics for the rate of escape, Green function and the Avez entropy of the random walk are also determined. Ergodic theory of the shift on the space of random walk paths on trees is the main tool; the key observation is that iterating the transformation induced from this shift to the subset of ‘exit points’ yields a nonintersecting path sampled from harmonic measure.

Towards a theory of negative dependence

The FKG theorem says that the positive lattice condition, an easily checkable hypothesis which holds for many natural families of events, implies positive association, a very useful property. Thus there is a natural and useful theory of positively dependent events. There is, as yet, no corresponding theory of negatively dependent events. There is, however, a need for such a theory. This paper, unfortunately, contains no substantial theorems. Its purpose is to present examples that motivate a need for such a theory, give plausibility arguments for the existence of such a theory, outline a few possible directions such a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.

Asymptotics of Multivariate Sequences II: Multiple Points of the Singular Variety

Let

Vertex-reinforced random walk

F(\b{z})=\sum_\b{r} a_\b{r}\b{z^r}

Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions

be a multivariate generating function that is meromorphic in some neighbourhood of the origin of

A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes

\mathbb{C}^d

Asymptotics of Multivariate Sequences

, and let

Moment conditions for a sequence with negative drift to be uniformly bounded in Lr

\sing

Network formation by reinforcement learning: the long and medium run

be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of