The two-parameter Poisson-Dirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to Vershik-Shmidt-Ignatov, are generalized to the two-parameter case. The size-biased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...

Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Delta-valued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i +x j at rate x i +x j . They showed that a version (X 1 (t); \Gamma1 ! t ! 1) of this process arises as a n !1 weak limit of the process started at time \Gamma 1 2 log n with n clusters of mass 1=n. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. We describe the distribution of X 1 (t) on \Delta at a fixed time t. We show that the size of the cluster containing a given atom, as a process in t, has a simple representation in terms of the stable subordinator of index 1=2. As t ! \Gamma1, we establish a Gaussian limit for (centered and norm...

Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...

Abstract not found

: A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P n V n = 1, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let V n be the length of the nth longest component interval of [0; 1]nZ, where Z is an a.s. non-empty random closed of (0; 1) of Lebesgue measure 0, and Z is self-similar, i.e. cZ has the same distribution as Z for every c ? 0. Then for 0 a ! b 1 the expected number of n's such that V n 2 (a; b) equals R b a v \Gamma1 F (dv) where the structural distribution F is identical to the distribution of 1 \Gamma sup(Z " [0; 1]). Then F (dv) = f(v)dv where (1 \Gamma v)f(v) is a decreasing function of v, and every such probability distribution F on [0; 1] can arise from this construction. Keywords: interval partition, zero set, excursion lengths, regenerative set, structural distribution. AMS subject classificat...

The infinitely divisible distributions on R + of random variables C t , S t and T t with Laplace transforms ` 1 cosh p 2 ' t ; / p 2 sinh p 2 ! t ; and / tanh p 2 p 2 ! t respectively are characterized for various t ? 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their L'evy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of C¹ and S³ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and ...

Abstract. We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola. The average cost of unsuccessful searches is considered too. 1.

The Joyal bijection between doubly-rooted trees and mappings can be lifted to a transformation on function space which takes tree-walks to mapping-walks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the 1994 Aldous-Pitman result on convergence of uniform random mapping walks to reflecting Brownian bridge, and extend this result to random p-mappings.

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where the distribution of M br+ 1 = sup 0t1 B br t is given by L'evy's formula P (M br+ 1 ? x) = e \Gamma2x 2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian bridge (jB br t j; 0 t 1) is given by a modification of the known `-function series for the density of M br 1 = sup 0t1 jB br t j. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a self-similar recurrent Markov process. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, selfsimilar recurrent Markov process, Bessel p...

Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursion lengths prior to T is the same as if T were a fixed time. It follows that the generalized arc-sine laws of Lamperti extend to such random times T . For some other random times T , absolute continuity relations are obtained which relate the law of the relative lengths at time T to the law at a fixed time. 1 Introduction Following Lamperti [10], Wendel [24], Kingman [7], Knight [8], PermanPitman -Yor [12, 13, 15], consider the sequence V 1 (T ) V 2 (T ) \Delta \Delta \Delta (1) of ranked lengths of component intervals of the set [0; T ]nZ, where T is a strictly positive random time, and Z is the zero set of a Markov process X started at zero, such as a Brownian motion or Bessel process,...