It is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $\alpha<1$ , and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line $(0,\infty)$ . The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.

In this paper we obtain a recursive formula for the density of the double-barrier Parisian stopping time. We present a probabilistic proof of the formula for the first few steps of the recursion, and then a formal proof using explicit Laplace inversions. These results provide an efficient computational method for pricing double-barrier Parisian options.

We study investment and disinvestment decisions in situations where there is a time lag d > 0 from the time t when the decision is taken to the time t + d when the decision is implemented. In this paper we apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay. In particular, we prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time. Relying on this result, we solve the constrained maximization problem, obtaining an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. We compare our results with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest, extending a result of Bar-Ilan and Strange (1996).

We study the following model for a phylogenetic tree on n extant species: the origin of the clade is a random time in the past whose (improper) distribution is uniform on (0,∞); thereafter, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on there being the prescribed number n of species at the present time. We study various mathematical properties of this model as n→∞: namely the time of origin and of the most recent common ancestor, the pattern of divergence times within lineage trees, the time series of the number of species, the total number of extinct species, the total number of species ancestral to the extant ones, and the ‘local’ structure of the tree itself. We emphasize several mathematical techniques: the association of walks with trees; a point process representation of lineage trees; and Brownian limits.

Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: a direct method based on Kac's formula for Brownian functionals and an indirect one based on a limit theorem for Galton–Watson trees.

In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

This paper is concerned with a random walk process in which and for i = 1, 2, ···, 2n. This process is called a Bernoulli excursion. The main object is to find the distribution, the moments, and the asymptotic distribution of the random variable ω n defined by . The results derived have various applications in the theory of probability, including random graphs, tournaments and order statistics.

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.

Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.

The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

The distribution of the maximum of the unsigned scaled Brownian excursion process and of a modification of that process are derived. These distributions are related to the one-dimensional Brownian bridge.