The beta-Bernoulli process provides a Bayesian nonparametric prior for models
involving collections of binary-valued features. A draw from the beta process
provides an infinite collection of probabilities in the unit interval, and a
draw from the Bernoulli process turns these into binary-valued features. Recent
work has shown how to derive stick-breaking representations for the beta
process, by analogy to Sethuraman's derivation of a stick-breaking
representation for the Dirichlet process.

This article contains both a point process and a sequential description of
the greatest convex minorant of Brownian motion on a finite interval. We use
these descriptions to provide new analysis of various features of the convex
minorant such as the set of times where the Brownian motion meets its minorant.
The equivalence of the these descriptions is non-trivial, which leads to many
interesting identities between quantities derived from our analysis. The
sequential description can be viewed as a Markov chain for which we derive some
fundamental properties.

We provide a representation of the maximal difference between a standard
Brownian bridge and its concave majorant on the unit interval, from which we
deduce expressions for the distribution and density functions and moments of
this difference. This maximal difference has an application in nonparametric
statistics where it arises in testing monotonicity of a density or regression
curve.

We use a natural ordered extension of the Chinese Restaurant Process to grow
a two-parameter family of binary self-similar continuum fragmentation trees. We
provide an explicit embedding of Ford's sequence of alpha model trees in the
continuum tree which we identified in a previous article as a distributional
scaling limit of Ford's trees. In general, the Markov branching trees induced
by the two-parameter growth rule are not sampling consistent, so the existence
of compact limiting trees cannot be deduced from previous work on the sampling
consistent case.