262,308 research outputs found
Convergence results for multivariate martingales
No full textWe present a new version of the Central Limit Theorem for multivariate martingales
An almost sure conditional convergence result and an application to a generalized Polya urn
No full textWe prove an almost sure conditional convergence result toward a Gaussian kernel and we apply it to a two-colors randomly reinforced urn
Asymptotic results for a generalized Pólya urn with “multi-updating” and applications to clinical trials
No full textIn this article, a new Pólya urn model is introduced and studied; in particular, a strong law of large numbers and two central limit theorems are proved. This urn generalizes a model studied in Berti et al. (2004), May et al. (2005), and in Crimaldi (2007), and it has natural applications in clinical trials. Indeed, the model includes both delayed and missing (or null) responses. Moreover, a connection with the conditional identity in distribution of Berti et al. (2004) is given
Central limit theorems for multicolor urns with dominated colors
No full textAn 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
Statistical test for an urn model with random multidrawing and random addition
No full textWe complete the study of the model introduced in Crimaldi et al., (2022). It is a two-color urn model with multiple drawing and random (non-balanced) time-dependent reinforcement matrix. The number of sampled balls at each time-step is random. We identify the exact rates at which the number of balls of each color grows to +infinity and define two strongly consistent estimators for the limiting reinforcement averages. Then we prove a Central Limit Theorem, which allows to design a statistical test for such averages.(c) 2023 Elsevier B.V. All rights reserved
Synchronization of Reinforced Stochastic Processes with a Network-based Interaction
No full textRandomly 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 Crimaldi et al. (2016, arXiv:1602.06217). 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
Interacting Reinforced Stochastic Processes: Statistical Inference based on the Weighted Empirical Means
No full textThis work deals with a system of interacting reinforced stochastic processes, where each process X-j = (X-n,X-j )(n), is located at a vertex j of a finite weighted directed graph, and it can be interpreted as the sequence of "actions" adopted by an agent j of the network. The interaction among the 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" Z(n,j) and on the inclinations Z(n,h), with h not equal j, of the other agents according to the entries of W. The best known example of reinforced stochastic process is the Polya urn.
The present paper focuses on the weighted empirical means N-n,N-j = Sigma(n)(k)(=1) q(n,k) X-k,X- j, since, for example, the current experience is more important than the past one in reinforced learning. Their almost sure synchronization and some central limit theorems in the sense of stable convergence are proven. The new approach with weighted means highlights the key points in proving some recent results for the personal inclinations Z(j) = (Z(n,j))(n) and for the empirical means (X) over bar (j) = (Sigma(n)(k)(=1) X-k,X- j/n)(n) given in recent papers (e.g. Aletti, Crimaldi and Ghiglietti (2019), Ann. Appl. Probab. 27 (2017) 3787-3844, Crimaldi et al. Stochastic Process. Appl. 129 (2019) 70-101). In fact, with a more sophisticated decomposition of the considered processes, we can understand how the different convergence rates of the involved stochastic processes combine. From an application point of view, we provide confidence intervals for the common limit inclination of the agents and a test statistics to make inference on the matrix W, based on the weighted empirical means. In particular, we answer a research question posed in Aletti, Crimaldi and Ghiglietti (2019)
A model for the Twitter sentiment curve
No full textTwitter is among the most used online platforms for the political communications, due to the concision of its messages (which is particularly suitable for political slogans) and the quick diffusion of messages. Especially when the argument stimulate the emotionality of users, the content on Twitter is shared with extreme speed and thus studying the tweet sentiment if of utmost importance to predict the evolution of the discussions and the register of the relative narratives. In this article, we present a model able to reproduce the dynamics of the sentiments of tweets related to specific topics and periods and to provide a prediction of the sentiment of the future posts based on the observed past. The model is a recent variant of the Polya urn, introduced and studied in Aletti and Crimaldi (2019, 2020), which is characterized by a "local" reinforcement, i.e. a reinforcement mechanism mainly based on the most recent observations, and by a random persistent fluctuation of the predictive mean. In particular, this latter feature is capable of capturing the trend fluctuations in the sentiment curve. While the proposed model is extremely general and may be also employed in other contexts, it has been tested on several Twitter data sets and demonstrated greater performances compared to the standard Polya urn model. Moreover, the different performances on different data sets highlight different emotional sensitivities respect to a public event
On the behavior of the conditional expectations in Skorohod representation theorem
No full textIn this paper we deal with the Skorohod representation of a given system of probability measures. More precisely, we give conditions for the existence of a Skorohod representation (X,(Xn)) with the following additional property: for each real number p⩾1 and each real random variable Z in Lp, the conditional expectation E[Z|Xn] converges in Lp to the conditional expectation E[Z|X
Twitter as an innovation process with damping effect
No full textIn the existing literature about innovation processes, the proposed models often satisfy the Heaps’ law, regarding the rate at which novelties appear, and the Zipf’s law, that states a power law behavior for the frequency distribution of the elements. However, there are empirical cases far from showing a pure power law behavior and such a deviation is mostly present for elements with high frequencies. We explain this phenomenon by means of a suitable “damping” effect in the probability of a repetition of an old element. We introduce an extremely general model, whose key element is the update function, that can be suitably chosen in order to reproduce the behaviour exhibited by the empirical data. In particular, we explicit the update function for some Twitter data sets and show great performances with respect to Heaps’ law and, above all, with respect to the fitting of the frequency-rank plots for low and high frequencies. Moreover, we also give other examples of update functions, that are able to reproduce the behaviors empirically observed in other contexts
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