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

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

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

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

Recruitment of Podosome Components Involved in the Remodelling of the Actin Cytoskeleton

Podosomes are dynamic, actin-rich structures that are involved in cell adhesion and extracellular matrix degradation; they are composed of a densely packed actin core surrounded by a ring structure made of components commonly found in focal adhesions. Podosome formation is characterized by the recruitment of AFAP-110, p190RhoGAP, and cortactin, which have specific roles in Src activation, local down-regulation of RhoA activity, and actin polymerization, respectively. However, the precise function of p190RhoGAP in podosome formation is not clear yet. By employing siRNA-mediated expression knockdown and expressing a catalytically inactive point mutant, I provide evidence that p190RhoGAP is required for podosome formation. It is well documented that Src-induced interaction of p190RhoGAP with p120RasGAP regulates p190RhoGAP activity and subcellular localization. In this thesis, I show that p190RhoGAP is constitutively associated in a complex with p120RasGAP in vascular smooth muscle cells, and that p120RasGAP translocates to podosomes upon PDBu stimulation. Nevertheless, siRNA-mediated knockdown of p120RasGAP expression does not impair p190RhoGAP recruitment or podosome formation, indicating that p120RasGAP is not essential for podosome formation. The molecular mechanism that underlies the specific recruitment of critical podosome components to sites of podosome formation remains unknown. The scaffold protein Tks5 is localized to podosomes in Src-transformed fibroblasts and in vascular smooth muscle cells, and may serve as a specific recruiting adapter for various components during podosome formation. I show here that induced mislocalization of Tks5 to the surface of mitochondria leads to a major subcellular redistribution of AFAP-110, p190RhoGAP, and cortactin, and to inhibition of podosome formation in vascular smooth muscle cells. Analysis of a series of similarly mistargeted deletion mutants of Tks5 indicates that the fifth SH3 domain is essential for this recruitment. A Tks5-GFP mutant lacking the PX domain also inhibits podosome formation and induces the redistribution of AFAP-110, p190RhoGAP, and cortactin to the perinuclear area. Together these findings demonstrate that Tks5 plays a central role in the recruitment of AFAP-110, p190RhoGAP, and cortactin to drive podosome formation. Evidence from osteoclasts and tumour cells cultured on different substrates indicates that the physical parameters of the underlying substrate influence the ability of cells to form podosomes or the related structures invadopodia. However, it is unclear how vascular smooth muscle cells respond to contact with different types of substrates. Thus, the last part of this thesis is dedicated to determine how podosome-forming vascular smooth muscle cells respond to alterations in the properties of the underlying substrate. I show here that A7r5 cells cultured on cross-linked gelatin degrade matrix by forming invadopodia-like structures. This is the first time that a cell type is reported to be capable of forming both podosomes and invadopodia in different conditions

Central limit theorems for multicolor urns with dominated colors

An urn contains balls of d≥2 colors. At each time n≥1, a ball is drawn and then replaced together with a random number of balls of the same color. Let A n = diag (An,1,…,An,d) be the n-th reinforce matrix. Assuming that EAn,j=EAn,1 for all n and j, a few central limit theorems (CLTs) are available for such urns. In real problems, however, it is more reasonable to assume that EA n,j = EA n,1 whenever  n ≥ 1  and  1 ≤ j ≤ d0 , liminfn EAn,1 > limsupn EAn,j whenever  j > d0 for some integer 1≤d0≤d. Under this condition, the usual weak limit theorems may fail, but it is still possible to prove the CLTs for some slightly different random quantities. These random quantities are obtained by neglecting dominated colors, i.e., colors from d0+1 to d, and they allow the same inference on the urn structure. The sequence (An : n ≥ 1) is independent but need not be identically distributed. Some statistical applications are given as well