Central limit theorems for a hypergeometric randomly reinforced urn

We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color given the past is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.Comment: 15 pages, submitted, Key-words: Central Limit Theorem; Polya urn; Randomly Reinforced Urn; Stable Convergenc

An almost sure conditional convergence result and an application to a generalized Polya urn

We prove an almost sure conditional convergence result toward a Gaussian kernel and we apply it to a two-colors randomly reinforced urn

Convergence results for conditional expectations

Let E,F be two Polish spaces and [Xn,Yn],[X,Y] random variables with values in E×F (not necessarily defined on the same probability space). We show some conditions which are sufficient in order to assure that, for each bounded continuous function f on E×F, the conditional expectation of f(Xn,Yn) given Yn converges in distribution to the conditional expectation of f(X,Y) given Y

A Network Model characterized by a Latent Attribute Structure with Competition

The quest for a model that is able to explain, describe, analyze and simulate real-world complex networks is of uttermost practical as well as theoretical interest. In this paper we introduce and study a network model that is based on a latent attribute structure: each node is characterized by a number of features and the probability of the existence of an edge between two nodes depends on the features they share. Features are chosen according to a process of Indian-Buffet type but with an additional random "fitness" parameter attached to each node, that determines its ability to transmit its own features to other nodes. As a consequence, a node's connectivity does not depend on its age alone, so also "young" nodes are able to compete and succeed in acquiring links. One of the advantages of our model for the latent bipartite "node-attribute" network is that it depends on few parameters with a straightforward interpretation. We provide some theoretical, as well experimental, results regarding the power-law behaviour of the model and the estimation of the parameters. By experimental data, we also show how the proposed model for the attribute structure naturally captures most local and global properties (e.g., degree distributions, connectivity and distance distributions) real networks exhibit. keyword: Complex network, social network, attribute matrix, Indian Buffet processComment: 34 pages, second version (date of the first version: July, 2014). Submitte

Rate of convergence of predictive distributions for dependent data

This paper deals with empirical processes of the type $$C_n(B)=\sqrt{n}\{\mu_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},$$ where $(X_n)$ is a sequence of random variables and $\mu_n=(1/n)\sum_{i=1}^n\delta_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ191 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotics for randomly reinforced urns with random barriers

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L<U\leq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\overset{a.s.}\longrightarrow Z$ for some random variable $Z$, and \begin{gather*} D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,\sigma^2)\quad\text{conditionally a.s.} \end{gather*} where $\sigma^2$ is a certain random variance. Almost sure conditional convergence means that \begin{gather*} P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)\overset{weakly}\longrightarrow\mathcal{N}(0,\,\sigma^2)\quad\text{a.s.} \end{gather*} where $P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)$ is a regular version of the conditional distribution of $D_n$ given the past $\mathcal{G}_n$. Thus, in particular, one obtains $D_n\longrightarrow\mathcal{N}(0,\sigma^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.Comment: 13 pages, submitte

The Rescaled Polya Urn: local reinforcement and chi-squared goodness of fit test

Motivated by recent studies of big samples, this work aims at constructing a parametric model which is characterized by the following features: (i) a "local" reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness of fit result. This triple purpose has been achieved by the introduction of a new variant of the Eggenberger-Polya urn, that we call the "Rescaled" Polya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited from the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.Comment: 28 pages, 1 figur