arXiv subjects

Manuel González-Navarrete

Publications and source records attributed to Manuel González-Navarrete.

16 recordsLinked to original sources

arXiv2025

A stochastic model for the diffusion of competing opinions with trend-following, opposition, and indifference

We study a stochastic model for the diffusion of competing opinions in a population composed of three types of agents: trend-followers, opposers, and indifferent individuals. The decision dynamics are driven by reinforcement mechanisms, modulated by a latent trend process, allowing us to capture realistic features such as amplification, resistance, and randomness in opinion formation. We derive explicit formulas for the finite-time moments of the opinion count vector and establish a set of strong asymptotic results, including laws of large numbers, central limit theorems, laws of the iterated logarithm, and almost sure convergence of empirical distributions. In particular, we show how early fluctuations can persist or vanish depending on the balance between reinforcement and opposition. Our analysis relies on martingale techniques and offers closed-form expressions for key quantities, providing both theoretical insights and tools for simulations or applications in social dynamics, marketing, or information diffusion.

math.PR↗

arXiv2025

Functional central limit theorem for dependent models with finite memory

We provide complementary results for a family of models with dependence on their previous $k$-sum. Using a martingale-based approach, we establish a functional central limit theorem and analyze the limiting behavior of the center of mass. Additionally, we explore the connection between our findings and the study of certain reinforced random walks in the literature

math.PR↗

arXiv2024

Shrinkage Estimators for Beta Regression Models

The beta regression model is a useful framework to model response variables that are rates or proportions, that is to say, response variables which are continuous and restricted to the interval (0,1). As with any other regression model, parameter estimates may be affected by collinearity or even perfect collinearity among the explanatory variables. To handle these situations shrinkage estimators are proposed. In particular we develop ridge regression and LASSO estimators from a penalized likelihood perspective with a logit link function. The properties of the resulting estimators are evaluated through a simulation study and a real data application

stat.ME↗

arXiv2024

Lasso regularization for mixture experiments with noise variables

We apply classical and Bayesian lasso regularizations to a family of models with the presence of mixture and process variables. We analyse the performance of these estimates with respect to ordinary least squares estimators by a simulation study and a real data application. Our results demonstrate the superior performance of Bayesian lasso, particularly via coordinate ascent variational inference, in terms of variable selection accuracy and response optimization.

stat.ME↗

arXiv2024

Complementary asymptotic analysis for a minimal random walk

We discuss a complementary asymptotic analysis of the so called minimal random walk. More precisely, we present a version of the almost sure central limit theorem as well as a generalization of the recently proposed quadratic strong laws. In addition, alternative demonstrations of the functional limit theorems will be supplied based on a Pólya urn scheme instead of a martingale approach.

math.PR↗

arXiv2024

A complete characterization of a correlated Bernoulli process

We present a complete characterization of the asymptotic behaviour of a correlated Bernoulli sequence { which depends on the parameter $θ\in [0,1]$. A martingale theory based approach will allow} us to prove versions of the law of large numbers, quadratic strong law, law of iterated logarithm, almost sure central limit theorem and functional central limit theorem, in the case $θ\le 1/2$. For $θ> 1/2$, we will obtain a strong convergence to a non-degenerated random variable, including a central limit theorem and a law of iterated logarithm for the fluctuations.

math.PR↗

arXiv2024

On the asymptotics of a lazy reinforced random walk

Based on a martingale theory approach, we present a complete characterization of the asymptotic behaviour of a lazy reinforced random walk (LRRW) which shows three different regimes (diffusive, critical and superdiffusive). This allows us to prove versions of the law of large numbers, the quadratic strong law, the law of iterated logarithm, the almost sure central limit theorem and the functional central limit theorem in the diffusive and critical regimes. In the superdiffusive regime we obtain a strong convergence to a random variable, including a central limit theorem and a law of iterated logarithm for the fluctuations.

math.PR↗

arXiv2021

Lack of phase transitions in staggered magnetic systems. A comparison of uniqueness criteria

We study a ferromagnetic Ising model with a staggered cell-board magnetic field previously proposed for image processing [Maruani et al., Markov Processes Relat. Fields 1 (1995) \cite{MPS}]. We complement previous results on the existence of phase transitions at low temperature [González-Navarrete et al., J. Stat. Phys. 162 (2016)] by determining bounds to the region of uniqueness of Gibbs measures. We establish sufficient rigorous uniqueness conditions derived from three different criteria: (1) Dobrushin criterion [Dobrushin, Theory Probab. Appl. 13 (1968)], (2) Disagreement percolation [van den Berg and Maes, Ann. Probab. 22 (1994)] and (3) Dobrushin-Shlosman criteria [Dobrushin and Shlosman, in Statistical Physics and Dynamical Systems. Rigorous Results. (1985)]. These conditions are subsequently solved numerically and the resulting uniqueness regions compared.

math-ph↗

arXiv2021

Reinforced random walks under memory lapses

We introduce a one-dimensional random walk, which at each step performs a reinforced dynamics with probability $θ$ and with probability $1 - θ$, the random walk performs a step independent of the past. We analyse its asymptotic behaviour, showing a law of large numbers and characterizing the diffusive and superdiffusive regions. We prove central limit theorems and law of iterated logarithm based on the martingale approach.

math.PR↗

arXiv2020

Multidimensional walks with random tendency

We introduce a multidimensional walk with memory and random tendency. The asymptotic behaviour is characterized, proving a law of large numbers and showing a phase transition from diffusive to superdiffusive regimes. In first case, we obtain a functional limit theorem to Gaussian vectors. In superdiffusive, we obtain strong convergence to a non-Gaussian random vector and characterize its moments.

math.PR↗

arXiv2020

The Monty Hall problem revisited

We propose a new approach to solve the classical Monty Hall problem in its general form. The solution is based on basic tools of probability theory, by defining three elementary events which decompose the sample space into a partition. The probabilities of each element of the partition allow us to compute the conditional and marginal probabilities of winning.

math.HO↗

arXiv2019

The diffusion of opposite opinions in a randomly biased environment

We propose a model for diffusion of two opposite opinions. Here, the decision to be taken by each individual is a random variable which depends on the tendency of the population, as well on its own trend characteristic. The influence of the population trend can be positive, negative or non-existent in a random form. We prove a phase transition in the behaviour of the proportion of each opinion. Namely, the mean square proportions are linear functions of time in the diffusive case, but are given by a power law in the superdiffusive regime.

math.PR↗

arXiv2019

Urn models with two types of strategies

We study an urn process containing red and blue balls and two different strategies to reinforce the urn. Namely, a generalized Pólya-type strategy versus an i.i.d. one. At each step, one of the two reinforcement strategies is chosen by flipping a coin. We study the asymptotic behaviour of this urn model, and prove a law of large numbers, a central limit theorem and a functional limit theorem for the proportion of balls into the urn. A phase transition is also stated.

math.PR↗

arXiv2018

Non-Markovian random walks with memory lapses

We propose an approach to construct Bernoulli trials $\{X_i, i\ge 1\}$ combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past $S_i = X_1 + \ldots + X_i$, {defines} a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials $\{Y_i, i\ge 1\}$, called {\it memory switch sequence}. We introduce the concept of {\it memory lapses property}, which {is} characterized by intervals of consecutive independent steps in BSRD. The main results include classical limit theorems for a class of linear BSRD. In particular, we obtain a central limit theorem for a class of BSRD which generalizes some previous results in literature. Along the paper, several examples of potential applications are provided.

math.PR↗

arXiv2016

Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module

We study an alternative approach to model the dynamical behaviors of biological feedback loop, that is, a type-dependent spin system, this class of stochastic models was introduced by Fernández et. al (2009), and are useful since take account to inherent variability of gene expression. We analyze a non-symmetric feedback module being an extension for the repressilator, the first synthetic biological oscillator, invented by Elowitz and Leibler (2000). We consider a mean-field dynamics for a type-dependent Ising model, and then study the empirical-magnetization vector representing concentration of molecules. We apply a convergence result from stochastic jump processes to deterministic trajectories and present a bifurcation analysis for the associated dynamical system. We show that non-symmetric module under study can exhibit very rich behaviours, including the empirical oscillations described by repressilator.

math.PR↗

arXiv2015

Phase transition in ferromagnetic Ising model with a cell-board external field

We show the presence of a first-order phase transition for a ferromagnetic Ising model on $\mathbb{Z}^2$ with a periodical external magnetic field. The external field takes two values $h$ and $-h$, where $h>0$. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides $L_1\times L_2$ sites, such that the total value of the external field is zero. The phase transition holds if $h<\frac{2J}{L_1}+ \frac{2J}{L_2}$, where $J$ is an interaction constant. We prove a first-order phase transition using the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate.

math-ph↗