Consider $q_n$ a random pointed quadrangulation chosen equally likely among
the pointed quadrangulations with $n$ faces. In this paper we show that, when
$n$ goes to $+\infty$, $q_n$ suitably normalized converges weakly in a certain
sense to a random limit object, which is continuous and compact, and that we
name the Brownian map. The same result is shown for a model of rooted
quadrangulations and for some models of rooted quadrangulations with random
edge lengths. A metric space of rooted (resp. pointed) abstract maps that
contains the model of discrete rooted (resp. pointed) quadrangulations and the
model of the Brownian map is defined. The weak convergences hold in these
metric spaces.Comment: Published at http://dx.doi.org/10.1214/009117906000000557 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
In this paper, we show a strong relation between the depth first processes associated to Galton-Watson trees with finite variance, conditioned by the total progeny: the depth first walk, the depth first queue process, the height process; a consequence is that these processes (suitably normalized) converge to the same Brownian excursion. This provides an alternative proof of Aldous' one of the convergence of the depth first walk to the Brownian excursion which does not use the existence of a limit tree. The methods that we introduce allow one to compute some functionals of trees or discrete excursions; for example, we compute the limit law of the process of the height of nodes with a given out-degree, and the process of the height of nodes, root of a given subtree.
The first aim of this paper is to establish the weak convergence rate of nonlinear two-time-scale stochastic approximation algorithms. Its second aim is to introduce the averaging principle in the context of two-time-scale stochastic approximation algorithms. We first define the notion of asymptotic efficiency in this framework, then introduce the averaged two-time-scale stochastic approximation algorithm, and finally establish its weak convergence rate. We show, in particular, that both components of the averaged two-time-scale stochastic approximation algorithm simultaneously converge at the optimal rate √ n.
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