Abstract. Randomly evolving systems composed by elements which interact among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon, that could be roughly defined as the tendency of different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement, that recently has been introduced in [12]. Generally speaking, by reinforcement we mean any mechanism for which the probability that a given event occurs has an increasing dependence on the number of times that events of the same type occurred in the past. The particularity of systems of such stochastic processes is that synchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit. We focus on the relationship between the topology of the network of the interactions and the long-time synchronization phenomenon. After proving the almost sure synchronization, we provide some CLTs in the sense of stable convergence that establish the convergence rates and the asymptotic distributions for both convergence to the common limit and synchronization. The obtained results lead to the construction of asymptotic confidence intervals for the limit random variable and of statistical tests to make inference on the topology of the network given the observation of the reinforced stochastic processes positioned at the vertices.

BackgroundThe relative risk of developing idiopathic PD is 1.5 times greater in men than in women, but an increased female prevalence in LRRK2-carriers has been described in the Ashkenazi Jewish population. We report an update about the frequency of major LRRK2 mutations in a large series of consecutive patients with Parkinson's disease (PD), including extensive characterization of clinical features. In particular, we investigated gender-related differences in motor and non-motor symptoms in the LRRK2 population.Methods2976 unrelated consecutive Italian patients with degenerative Parkinsonism were screened for mutations on exon 41 (G2019S, I2020T) and a subgroup of 1190 patients for mutations on exon 31 (R1441C/G/H). Demographic and clinical features were compared between LRRK2-carriers and non-carriers, and between male and female LRRK2 mutation carriers.ResultsLRRK2 mutations were identified in 40 of 2523 PD patients (1.6%) and not in other primary parkinsonian syndromes. No major clinical differences were found between LRRK2-carriers and non-carriers. We found a novel I2020L missense variant, predicted to be pathogenic. Female gender was more common amongst carriers than non-carriers (57% vs. 40%; p = 0.01), without any gender-related difference in clinical features. Family history of PD was more common in women in the whole PD group, regardless of their LRRK2 status.ConclusionsPD patients with LRRK2 mutations are more likely to be women, suggesting a stronger genetic load compared to idiopathic PD. Further studies are needed to elucidate whether there is a different effect of gender on the balance between genetic and environmental factors in the pathogenesis of PD.

This work deals with systems of interacting reinforced stochastic processes, where each process X j = (Xn,j )n is located at a vertex j of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent j of the network. The interaction among the evolving dynamics of these processes depends on the weighted adjacency matrix W associated to the underlying graph: indeed, the probability that an agent j chooses a certain action depends on its personal "inclination" Zn,j and on the inclinations Z n,h , with h = j, of the other agents according to the elements of W .Asymptotic results for the stochastic processes of the personal inclinations Z j = (Zn,j )n have been subject of studies in recent papers (e.g. [2,21]); while the asymptotic behavior of quantities based on the stochastic processes X j of the actions has never been studied yet. In this paper, we fill this gap by characterizing the asymptotic behavior of the empirical means Nn,j = n k=1 X k,j /n, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. Moreover, we discuss some statistical applications of these convergence results concerning confidence intervals for the random limit toward which all the processes of the system converge and tools to make inference on the matrix W .

This paper proposes a clustering procedure for samples of multivariate functions in (L 2 (I)) J , with J ≥ 1. This method is based on a k-means algorithm in which the distance between the curves is measured with a metrics that generalizes the Mahalanobis distance in Hilbert spaces, considering the correlation and the variability along all the components of the functional data. The proposed procedure has been studied in simulation and compared with the k-means based on other distances typically adopted for clustering multivariate functional data. In these simulations, it is shown that the k-means algorithm with the generalized Mahalanobis distance provides the best clustering performances, both in terms of mean and standard deviation of the number of misclassified curves. Finally, the proposed method has been applied to two real cases studies, concerning ECG signals and growth curves, where the results obtained in simulation are confirmed and strengthened.

Abstract. We consider systems of interacting Generalized Friedman's Urns (GFUs) having irreducible mean replacement matrices. The interaction is modeled through the probability to sample the colors from each urn, that is defined as convex combination of the urn proportions in the system. From the weights of these combinations we individuate subsystems of urns evolving with different behaviors. We provide a complete description of the asymptotic properties of urn proportions in each subsystem by establishing limiting proportions, convergence rates and Central Limit Theorems. The main proofs are based on a detailed eigenanalysis and stochastic approximation techniques.Keywords. Interacting systems; Urn models; Strong Consistency; Central Limit Theorems; Stochastic approximation.2010 MSC classification: 60K35; 62L20; 60F05; 60F15. IntroductionThe stochastic evolution of systems composed by elements which interact among each other has always been of great interest in several areas of application, e.g. in medicine a tumor growth is the evolution of a system of interacting cells [35], in socio-economics and life sciences a collective phenomenon reflects the result of the interactions among the individuals [27], in physics the concentration of certain molecules within cells varies over time due to interactions between different cells [31]. In the last decade several models have been proposed in which the elements of the system are represented by urns containing balls of different colors, in which the urn proportions reflect the status of the elements, and the evolution of the system is established by studying the dynamics at discrete times of this collection of dependent urn processes. The main reason of this popularity is concerned with the urn dynamics, which is (i) suitable to describe random phenomena in different scientific fields (see e.g. [21]), (ii) flexible to cover a wide range of possible asymptotic behaviors, (iii) intuitive and easy to be implemented in several fields of application.The dynamics of a single urn typically consists in a sequential repetition of a sampling phase, when a ball is sampled from the urn, and a replacement phase, when a certain quantity of balls is replaced in the urn. The basic model is the Pólya's urn proposed in [16]: from an urn containing balls of two colors, balls are sequentially sampled and then replaced in the urn with a new ball of the same color. This updating scheme is then iterated generating a sequence of urn proportions whose almost sure limit is random and Beta distributed. Starting from this simple model, several interesting variations have been suggested by considering different distributions in the sampling phase, e.g. [19,20], or in the replacement phase, e.g. [3,18,30]. In a general K-colors urn model, the color sampled at time n is usually represented by a vector X n such that X i,n = 1 when the sampled color is i ∈ {1, . . ., K}, X i,n = 0 otherwise; the quantities of balls replaced in the urn at time n are typically defined by a matrix D n such that D ki,n i...