arXiv subjects

Rodrigo Lambert

Publications and source records attributed to Rodrigo Lambert.

8 recordsLinked to original sources

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 $\theta \in [0,1]$. 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 $\theta \le 1/2$. For $\theta > 1/2$, 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↗

arXiv2019

Matching strings in encoded sequences

We investigate the longest common substring problem for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the R\'enyi entropy of the source. We apply this result to the zero-inflated contamination model and the stochastic scrabble. In the case of dynamical systems, this problem is equivalent to the shortest distance between two observed orbits and its limiting relationship with the correlation dimension of the pushforward measure. An extension to the shortest distance between orbits for random dynamical systems is also provided.

math.PR↗

arXiv2018

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↗

arXiv2017

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 P\'olya-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↗

arXiv2016

From the divergence between two measures to the shortest path between two observables

We consider two independent and stationary measures over $\chi^\mathbb{N}$, where $\chi$ finite or countable alphabet. For each pair of $n$-strings in the product space we define $T_n^{(2)}$ as the length of the shortest path connecting one string to the other where the paths are generating by the underlying dynamics of the measure. For ergodic measures with positive entropy we prove that, for almost every pair of realizations $(x,y)$, $T^{(2)}_n/n$ concentrates in one, as $n$ diverges. Under mild extra conditions we prove a large deviation principle. This principle is linked to a quantity that compute the similarity between the two measures that we also introduce. We further prove its existence and other properties. We also show that the fluctuations of $T_n^{(2)}$ converge (only) in distribution to a non-degenerated distribution. Several examples are provided for all results.

math.PR↗

arXiv2016

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↗

arXiv2011

The distribution of the overlapping function

We consider the set of finite sequences of length n over a finite or countable alphabet C. We consider the function which associate each given sequence with the size of the maximum overlap with a (shifted) copy of itself. We compute the exact distribution and the limiting distribution of this function when the sequence is chosen according to a product measure with marginals identically distributed. We give a point-wise upper bound for the velocity of this convergence. Our results holds for a finite or countable alphabet. The non-parametric distribution is related to the prime decomposition of positive integers. We illustrate with some examples.

math.PR↗