On the profile of random trees

Abstract

Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree may be described by the number of nodes or the number of leaves in layer t √ n, respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic functions which are obtained by means of generating functions arising from the combinatorial setup and complex contour integration. Besides, an integral representation for the two dimensional density of Brownian excursion local time is derived.

Random cutting and records in deterministic and random trees

“…The profile and the width can be treated similarly [1,3,15], but the limits will be described by the local time of the Brownian excursion; this was extended by [10] to include joint distribution with the height. Chassaing, Marckert, and Yor [10] further gave a second proof using instead the breadth-first walk (see below), which proves…”

Profiles of random trees: Plane‐oriented recursive trees

“…We next consider the covariance of two level sizes. Define with the coefficients given by (16). Note that…”

“…By singularity analysis, we have, by choosing a suitable Hankel contour H (see [16] for similar details),…”

On a conditional invariance principle for a critical Galton–Watson branching process

“…For arbitrary lattice zero-drift random walks this relation is not proved yet, but for a simple random walk it follows from Drmota and Gittenberger's result for random trees [8]. For arbitrary lattice zero-drift random walks this relation is not proved yet, but for a simple random walk it follows from Drmota and Gittenberger's result for random trees [8].…”

“…It is known (see, e.g., [8]) that one-dimensional distributions of the local time of a Brownian excursion have an atom at zero and are absolutely continuous on (0, ∞). Thus, the finite-dimensional distribution functions in (21) and (22) converge for any positive values of the arguments.…”