Conditional limit theorems for random walks and convergence of random trees I

We start with a short and clearly incomplete review of conditional functional limit theorems for random walks. In particular, we consider a Brownian excursion, which appears as one of the limit processes, and give it a convenient “non-conditional” characterization in terms of a standard Brownian motion.
The second part of the talk is on random trees. A certain straightforward bijection maps the set of all rooted trees with $n$ edges into so-called Dyck paths of length $2n$. Assuming that the elements of these sets are equally likely, we observe that such random trees could be coded by positive excursions of length $2n$ of a simple random walk. The limit of the later is a Brownian excursion, and it is natural to assume that it codes a certain continuous random tree (CRT). We define this object called Aldous' CRT, and discuss, subject to time constraints, in which sense it is the weak limit of discrete random trees.