Results 1  10
of
14
The Bernoulli sieve revisited
, 2008
"... We consider an occupancy scheme in which ‘balls’ are identified with n points sampled from the standard exponential distribution, while the role of ‘boxes’ is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behaviour of five ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
We consider an occupancy scheme in which ‘balls’ are identified with n points sampled from the standard exponential distribution, while the role of ‘boxes’ is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behaviour of five quantities: the index K ∗ n of the last occupied box, the number Kn of occupied boxes, the number Kn,0 of empty boxes whose index is at most K ∗ n, the index Wn of the first empty box and the number of balls Zn in the last occupied box. It is shown that the limiting distribution of properly scaled and centered K ∗ n coincides with that of the number of renewals not exceeding log n. A similar result is shown for Kn and Wn under a side condition that prevents occurrence of very small boxes. The condition also ensures that Kn,0 converges in distribution. Limiting results for Zn are established under an assumption of regular variation.
ON THE EXTENDED MORAN MODEL AND ITS RELATION TO COALESCENTS WITH MULTIPLE COLLISIONS
, 2011
"... We study the asymptotics of the extended Moran model as the total population size N tends to infinity. Two convergence results are provided, the first result leading to discretetime limiting coalescent processes and the second result leading to continuoustime limiting coalescent processes. The lim ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
We study the asymptotics of the extended Moran model as the total population size N tends to infinity. Two convergence results are provided, the first result leading to discretetime limiting coalescent processes and the second result leading to continuoustime limiting coalescent processes. The limiting coalescent processes allow for multiple mergers of ancestral lineages (Λcoalescent). It is furthermore verified that any continuous time Λcoalescent (with Λ any probability distribution) can arise in the limit. Typical examples of extended Moran models are discussed, with an emphasis on models being in the domain of attraction of beta coalescents or Λcoalescents with Λ being log infinitely divisible.
On the length of an external branch in the betacoalescents
, 2012
"... In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of collisions which occur in the ncoalescent until the end of the chosen external branch, and for the block counting process associated with the ncoalescent.
On the number of zero increments of random walks with a barrier
"... This is the second submission of this document. ..."
(Show Context)
A CONSTRUCTION OF A βCOALESCENT VIA THE PRUNING OF BINARY TREES
"... Abstract. Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β ( 3 1,)coalescent process. We also use the 2 2 continuous analogue of this construction, i.e. a pruning procedure on Aldous’s continuum random tree, to construct a co ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β ( 3 1,)coalescent process. We also use the 2 2 continuous analogue of this construction, i.e. a pruning procedure on Aldous’s continuum random tree, to construct a continuous state space process that has the same structure as the βcoalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event. hal00711518, version 2 9 Nov 2012 1.
On the asymptotics of moments of linear random recurrences
 Theory Stoch. Proc
"... ar ..."
(Show Context)
On the number of empty boxes in the Bernoulli sieve I
 Stochastics
, 2013
"... ar ..."
(Show Context)