arXiv subjects

Sandro Gallo

Publications and source records attributed to Sandro Gallo.

At least 19 recordsLinked to original sources

arXiv2026

Immigration Processes with Binomial Catastrophes and Random Survival Parameters

We consider immigration processes with binomial catastrophes and random survival parameters. Two sources of randomness are analyzed. In the first model, the survival parameter is independently resampled at each catastrophe. In the second model, individuals are assigned independent survival parameters at birth, which remain fixed over time. We show that the first model exhibits almost sure extinction, as in the classical case with a fixed survival parameter. In contrast, the second model exhibits a phase transition, admitting survival with positive probability, depending on the distribution of individual survival parameters. We provide explicit formulas for both the survival probability and the expected time to extinction. Finally, our proofs establish a novel methodological bridge with the Firework process, unifying population dynamics with spatial models of information spreading.

math.PR↗

arXiv2025

The critical curve of long-range percolation on oriented trees

We consider a long-range percolation model on homogeneous oriented trees with several lengths. We obtain the critical surface as the set of zeros of a specific polynomial with coefficients depending explicitly on the lengths and the degree of the tree. Restricting to the case of two lengths, we obtain new bounds on the critical parameters, monotonicity properties, as well as continuity of the critical curve, plus some partial results concerning its convexity. Our proofs rely on the study of the properties of the characteristic polynomial of the transition matrix of a multi-step Markov chain related to the model.

math.PR↗

arXiv2025

Uniqueness of stationary compatible probability measures for chains of infinite order with forbidden transitions

In this paper, we consider chains of infinite order on countable state spaces with prohibited transitions. We give a set of sufficient conditions on the structure of the probability kernels of the chains to have at most one stationary probability measure compatible with the kernel. Our main result extends the uniqueness $\ell^2$ criterion from Johansson and \"Oberg (2003) which was obtained for strongly non-null chains. A particular attention is given to concrete examples, illustrating the main theorem and its corollaries, with comparison to results of the existing literature.

math.PR↗

arXiv2023

Fluctuations of the occupation density for a parking process

Consider the following simple parking process on $\Lambda_n := \{-n, \ldots, n\}^d,d\ge1$: at each step, a site $i$ is chosen at random in $\Lambda_n$ and if $i$ and all its nearest neighbor sites are empty, $i$ is occupied. Once occupied, a site remains so forever. The process continues until all sites in $\Lambda_n$ are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of $\Lambda_n$ is called the jamming limit and is denoted by $X_{\Lambda_n}$. Ritchie (2006) constructed a stationary random field on $\mathbb Z^d$ obtained as a (thermodynamic) limit of the $X_{\Lambda_n}$'s as $n$ tends to infinity. As a consequence of his construction, he proved a strong law of large numbers for the proportion of occupied sites in the box $\Lambda_n$ for the random field $X$. Here we prove the central limit theorem, the law of iterated logarithm, and a gaussian concentration inequality for the same statistics. A particular attention will be given to the case $d=1$, in which we also obtain new asymptotic properties for the sequence $X_{\Lambda_n},n\ge1$ as well as a new proof to the closed-form formula for the occupation density of the parking process.

math.PR↗

arXiv2023

Consistent model selection for the Degree Corrected Stochastic Blockmodel

The Degree Corrected Stochastic Block Model (DCSBM) was introduced by \cite{karrer2011stochastic} as a generalization of the stochastic block model in which vertices of the same community are allowed to have distinct degree distributions. On the modelling side, this variability makes the DCSBM more suitable for real life complex networks. On the statistical side, it is more challenging due to the large number of parameters when dealing with community detection. In this paper we prove that the penalized marginal likelihood estimator is strongly consistent for the estimation of the number of communities. We consider \emph{dense} or \emph{semi-sparse} random networks, and our estimator is \emph{unbounded}, in the sense that the number of communities $k$ considered can be as big as $n$, the number of nodes in the network.

math.ST↗

arXiv2022

Critical parameter of the frog model on homogeneous trees with geometric lifetime

We consider the frog model with geometric lifetime (parameter $1-p$) on homogeneous trees of dimension $d$. In 2002, \cite{alves2002-2} proved that there exists a critical lifetime parameter $p_c\in(0,1)$ above which infinitely many frogs are activated with positive probability, and they gave lower and upper bounds for $p_c$. Since then, the literature on this model focussed on refinements of the upper bound. In the present paper we improve the bounds for $p_c$ \emph{on both sides}. We also provide a discussion comparing the bounds of the literature and their proofs. Our proofs are based on coupling.

math.PR↗

arXiv2021

Number of visits in arbitrary sets for $ϕ$-mixing dynamics

It is well-known that, for sufficiently mixing dynamical systems, the number of visits to balls and cylinders of vanishing measure is approximately Poisson compound distributed in the Kac scaling. Here we extend this kind of results when the target set is an arbitrary set with vanishing measure in the case of $ϕ$-mixing systems. The error of approximation in total variation is derived using Stein-Chen method. An important part of the paper is dedicated to examples to illustrate the assumptions, as well as applications to temporal synchronisation of $g$-measures

math.DS↗

arXiv2021

Potential well in Poincaré recurrence

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes with two new results: (1) for $ϕ$-mixing and $ψ$-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well.

math.PR↗

arXiv2020

Non-regular g-measures and variable length memory chains

It is well-known that there always exists at least one stationary measure compatible with a continuous g-function g. Here we prove that if the set of discontinuities of the g-function g has null measure under a candidate measure obtained by some asymptotic procedure, then this candidate measure is compatible with g. We explore several implications of this result, and discuss comparisons with the literature concerning assumptions and examples. Important part of the paper is concerned with the case of variable length memory chains, for which we obtain existence, uniqueness and weak-Bernoullicity (or $β$-mixing) under new assumptions. These results are specially designed for variable length memory models, and do not require vanishing uniform variation. We also provide a further discussion on some related notions, such as random context processes, non-essential discontinuities, and finally an example of everywhere discontinuous stationary measure.

math.PR↗

arXiv2018

Frog models on trees through renewal theory

This paper studies a class of growing systems of random walks on regular trees, known as \emph{frog models with geometric lifetime} in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improve the bounds of the literature for the critical parameter of a percolation model on trees called \emph{cone percolation}

math.PR↗

arXiv2018

Discrete one-dimensional coverage process on a renewal process

We consider the {following} coverage model on $\mathbb{N}$. For each site $i\in \mathbb{N}$ we associate a pair $(ξ_i, R_i)$ where $\{ξ_0, ξ_1, \ldots \}$ is a 1-dimensional {undelayed} discrete renewal point process and $\{R_0,R_1,\ldots\}$ is an i.i.d. sequence of $\mathbb{N}$-valued random variables. At each site where $ξ_i=1$ we start an interval of length $R_i$. Coverage occurs if every site of $\mathbb{N}$ is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.

math.PR↗

arXiv2015

Characterization of the stability of chains associated with $g$-measures

In this paper we introduce a notion of asymptotic stability of a probability kernel, which we call dynamic uniqueness. We say that a kernel exhibits dynamic uniqueness if all the stochastic chains starting from a fixed past coincide on the future tail $σ$-algebra. We prove that the dynamic uniqueness is generally stronger than the usual notion of uniqueness for $g$-measures. Our main result shows that dynamic uniqueness is equivalent to the weak-$\ell^2$ summability condition on the kernel. This generalizes and strengthens the Johansson-Öberg $\ell^2$ criterion for uniqueness of $g$-measures. Finally, among other things, we prove that the weak-$\ell^2$ criterion implies $β$-mixing of the unique $g$-measure compatible with a regular kernel improving several results in the literature.

math.PR↗

arXiv2015

Nonparametric statistical inference for the context tree of a stationary ergodic process

We consider the problem of estimating the context tree of a stationary ergodic process with finite alphabet without imposing additional conditions on the process. As a starting point we introduce a Hamming metric in the space of irreducible context trees and we use the properties of the weak topology in the space of ergodic stationary processes to prove that if the Hamming metric is unbounded, there exist no consistent estimators for the context tree. Even in the bounded case we show that there exist no two-sided confidence bounds. However we prove that one-sided inference is possible in this general setting and we construct a consistent estimator that is a lower bound for the context tree of the process with an explicit formula for the coverage probability. We develop an efficient algorithm to compute the lower bound and we apply the method to test a linguistic hypothesis about the context tree of codified written texts in European Portuguese.

math.ST↗

arXiv2014

Explicit estimates in the Bramson-Kalikow model

The aim of the present article is to explicitly compute parameters for which the Bramson-Kalikow model exhibits phase-transition. The main ingredient of the proof is a simple new criterion for non-uniqueness of $g$-measures. We show that the existence of multiple $g$-measures compatible with a function $g$ can be proved by estimating the $\bar{d}$-distances between some suitably chosen Markov chains. The method is optimal for the important class of binary regular attractive functions, which includes the Bramson-Kalikow model.

math.PR↗

arXiv2014

Continuity properties of a factor of Markov chains

Starting from a Markov chain with a finite alphabet, we consider the chain obtained when all but one symbol are undistinguishable for the practitioner. We study necessary and sufficient conditions for this chain to have continuous transition probabilities with respect to the past.

math.PR↗

arXiv2014

Rumor processes on $\N$ and discrete renewal processes

We study two rumor processes on $\N$, the dynamics of which are related to an SI epidemic model with long range transmission. Both models start with one spreader at site $0$ and ignorants at all the other sites of $\N$, but differ by the transmission mechanism. In one model, the spreaders transmit the information within a random distance on their right, and in the other the ignorants take the information from a spreader within a random distance on their left. We obtain the probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our proofs is to show that, in each model, the position of the spreaders on $\N$ can be related to a suitably chosen discrete renewal process.

math.PR↗

arXiv2013

Perfect simulation for locally continuous chains of infinite order

We establish sufficient conditions for perfect simulation of chains of infinite order on a countable alphabet. The new assumption, localized continuity, is formalized with the help of the notion of context trees, and includes the traditional continuous case, probabilistic context trees and discontinuous kernels. Since our assumptions are more refined than uniform continuity, our algorithms perfectly simulate continuous chains faster than the existing algorithms of the literature. We provide several illustrative examples.

math.PR↗

arXiv2012

Attractive regular stochastic chains: perfect simulation and phase transition

We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an i.i.d. process with countable alphabet, (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson-Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.

math.PR↗