Searcharxiv⌕ Search

arXiv subjects

Miguel González

Publications and source records attributed to Miguel González.

At least 19 recordsLinked to original sources
arXiv2026

Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles

Let \(\{X_n:n\ge0\}\) be a controlled branching process with controls \(ϕ_n\), and let \(N_n=ϕ_n(X_n)\) be its progenitor count. We study large deviations of \(X_{n+1}/N_n\), \(N_n/X_n\), and \(X_{n+1}/X_n\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur--Rao--Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental--boundary--persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact \(n^{-1/2}\)-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.

math.PR↗
arXiv2026

A scaling limit theorem for controlled branching processes with a size-divisible term

This paper establishes general sufficient conditions for a sequence of controlled branching processes to converge weakly on the Skorokhod space. We focus on a class of control mechanisms that extend previous results by decomposing those random variables into the sum of two independent components: an immigration term, which depends on the current population size, and a size-divisible term, which can be expressed as the sum of random contributions from each individual. This extension allows us to capture a broad range of control functions including Poisson, binomial, and negative binomial distributions, commonly used in the literature. The assumptions are formulated in terms of probability generating functions of the offspring and control laws, distinguishing in this latter between the immigration and the size-divisible parts. The limit process is shown to be a continuous-state branching process with dependent immigration. The proof essentially relies on tightness arguments and the identification of a martingale problem. We also identify the special case in which the limit reduces to a classical Feller branching diffusion with immigration.

math.PR↗
arXiv2026

Very stable parabolic $G$-Higgs bundles and affine flag varieties

Let $G$ be a semisimple complex algebraic group. We study the $\mathbb C^\times$-action on the moduli space of strongly parabolic $G$-Higgs bundles over a punctured curve with weights in the interior of the standard Weyl alcove. We describe its fixed points and, for the components with generically regular Higgs field, we classify the fixed points that have closed Bialynicki-Birula upward flows, called very stable. The classification is stated in terms of a naturally associated divisor valued in the extended affine Weyl group of $G$ which encodes the information of the fixed point. Under this point of view, very stable fixed points naturally coincide with divisors where the coefficients are minimal under the Bruhat order. Furthermore, when the divisor is supported at the parabolic punctures, the corresponding very stable upward flow is a section of the Hitchin map. For these upward flows, we propose a conjectural mirror line bundle over the dual moduli space, and verify its validity for $G=\mathrm{PGL}_n(\mathbb{C})$.

math.AG↗
arXiv2026

Particle filtering methods for partially observed branching processes

This paper focuses on the estimation of partially observed branching processes. First, the estimators from a frequentist perspective proposed in the literature are reviewed. The main objective of this paper is to present computational tools based on sequential Monte Carlo methods to perform Bayesian inference for these processes. In particular, the Liu-West particle filter is applied to perform Bayesian estimation of the parameters of interest for an epidemic model fitted by a partially observed branching process. As application, the example given in [8] is revisited and extended.

stat.CO↗
arXiv2026

Cyclic Higgs bundles and the Toledo invariant

Let $G$ be a complex semisimple Lie group and $\mathfrak g$ its Lie algebra. In this paper, we study a special class of cyclic Higgs bundles constructed from a $\mathbb Z$-grading $\mathfrak g = \bigoplus_{j=1-m}^{m-1}\mathfrak g_j$ by using the natural representation $G_0 \to GL(\mathfrak g_1 \oplus \mathfrak g_{1-m})$, where $G_0 \le G$ is the connected subgroup corresponding to $\mathfrak g_0$. The resulting Higgs pairs include $G^{\mathbb R}$-Higgs bundles for $G^{\mathbb R} \le G$ a real form of Hermitian type (in the case $m=2$) and fixed points of the ${\mathbb C}^*$-action on $G$-Higgs bundles (in the case where the Higgs field vanishes along $\mathfrak g_{1-m}$). In both of these situations a topological invariant with interesting properties, known as the Toledo invariant, has been defined and studied in the literature. This paper generalises its definition and properties to the case of arbitrary $(G_0,\mathfrak g_1 \oplus \mathfrak g_{1-m})$-Higgs pairs, which give rise to families of cyclic Higgs bundles. The results are applied to the example with $m=3$ that arises from the theory of quaternion-Kähler symmetric spaces.

math.AG↗
arXiv2025

La Leaderboard: A Large Language Model Leaderboard for Spanish Varieties and Languages of Spain and Latin America

Leaderboards showcase the current capabilities and limitations of Large Language Models (LLMs). To motivate the development of LLMs that represent the linguistic and cultural diversity of the Spanish-speaking community, we present La Leaderboard, the first open-source leaderboard to evaluate generative LLMs in languages and language varieties of Spain and Latin America. La Leaderboard is a community-driven project that aims to establish an evaluation standard for everyone interested in developing LLMs for the Spanish-speaking community. This initial version combines 66 datasets in Basque, Catalan, Galician, and different Spanish varieties, showcasing the evaluation results of 50 models. To encourage community-driven development of leaderboards in other languages, we explain our methodology, including guidance on selecting the most suitable evaluation setup for each downstream task. In particular, we provide a rationale for using fewer few-shot examples than typically found in the literature, aiming to reduce environmental impact and facilitate access to reproducible results for a broader research community.

cs.CL↗
arXiv2025

Psycholinguistic Word Features: a New Approach for the Evaluation of LLMs Alignment with Humans

The evaluation of LLMs has so far focused primarily on how well they can perform different tasks such as reasoning, question-answering, paraphrasing, or translating. For most of these tasks, performance can be measured with objective metrics, such as the number of correct answers. However, other language features are not easily quantified. For example, arousal, concreteness, or gender associated with a given word, as well as the extent to which we experience words with senses and relate them to a specific sense. Those features have been studied for many years by psycholinguistics, conducting large-scale experiments with humans to produce ratings for thousands of words. This opens an opportunity to evaluate how well LLMs align with human ratings on these word features, taking advantage of existing studies that cover many different language features in a large number of words. In this paper, we evaluate the alignment of a representative group of LLMs with human ratings on two psycholinguistic datasets: the Glasgow and Lancaster norms. These datasets cover thirteen features over thousands of words. The results show that alignment is \textcolor{black}{generally} better in the Glasgow norms evaluated (arousal, valence, dominance, concreteness, imageability, familiarity, and gender) than on the Lancaster norms evaluated (introceptive, gustatory, olfactory, haptic, auditory, and visual). This suggests a potential limitation of current LLMs in aligning with human sensory associations for words, which may be due to their lack of embodied cognition present in humans and illustrates the usefulness of evaluating LLMs with psycholinguistic datasets.

cs.CL↗
arXiv2025

Recursive InPainting (RIP): how much information is lost under recursive inferences?

The rapid adoption of generative artificial intelligence (AI) is accelerating content creation and modification. For example, variations of a given content, be it text or images, can be created almost instantly and at a low cost. This will soon lead to the majority of text and images being created directly by AI models or by humans assisted by AI. This poses new risks; for example, AI-generated content may be used to train newer AI models and degrade their performance, or information may be lost in the transformations made by AI which could occur when the same content is processed over and over again by AI tools. An example of AI image modifications is inpainting in which an AI model completes missing fragments of an image. The incorporation of inpainting tools into photo editing programs promotes their adoption and encourages their recursive use to modify images. Inpainting can be applied recursively, starting from an image, removing some parts, applying inpainting to reconstruct the image, revising it, and then starting the inpainting process again on the reconstructed image, etc. This paper presents an empirical evaluation of recursive inpainting when using one of the most widely used image models: Stable Diffusion. The inpainting process is applied by randomly selecting a fragment of the image, reconstructing it, selecting another fragment, and repeating the process a predefined number of iterations. The images used in the experiments are taken from a publicly available art data set and correspond to different styles and historical periods. Additionally, photographs are also evaluated as a reference. The modified images are compared with the original ones by both using quantitative metrics and performing a qualitative analysis. The results show that recursive inpainting in some cases modifies the image so that it still resembles the original one while in others leads to degeneration.

cs.CV↗
arXiv2025

Have Multimodal Large Language Models (MLLMs) Really Learned to Tell the Time on Analog Clocks?

Multimodal Large Language Models which can answer complex questions on an image struggle to tell the time on analog clocks. This is probably due to the lack of images with clocks at different times in their training set. In this work we explore this issue with one of the latest MLLMs: GPT-4.1 to understand why MLLMs fail to tell the time and whether fine-tuning can solve the problem. The results show how models are making progress in reading the time on analog clocks. But have they really learned to do it, or have they only learned patterns in their training datasets? In this work we put the models to the test with different clocks to illustrate the limitations of MLLMs to abstract and generalize.

cs.CL↗
arXiv2025

Very stable regular $G$-Higgs bundles

We give a classification of very stable $G$-Higgs bundles in the generically regular Higgs field case for $G$ an arbitrary connected semisimple complex group. This extends the classification for $G=\mathrm{GL}_n(\mathbb C)$ and fixed point type $(1,1,\dots,1)$ given by Hausel and Hitchin.

math.AG↗
arXiv2024

Open Conversational LLMs do not know most Spanish words

The growing interest in Large Language Models (LLMs) and in particular in conversational models with which users can interact has led to the development of a large number of open-source chat LLMs. These models are evaluated on a wide range of benchmarks to assess their capabilities in answering questions or solving problems on almost any possible topic or to test their ability to reason or interpret texts. Instead, the evaluation of the knowledge that these models have of the languages has received much less attention. For example, the words that they can recognize and use in different languages. In this paper, we evaluate the knowledge that open-source chat LLMs have of Spanish words by testing a sample of words in a reference dictionary. The results show that open-source chat LLMs produce incorrect meanings for an important fraction of the words and are not able to use most of the words correctly to write sentences with context. These results show how Spanish is left behind in the open-source LLM race and highlight the need to push for linguistic fairness in conversational LLMs ensuring that they provide similar performance across languages.

cs.CL↗
arXiv2024

Critical Multitype Branching Processes with Random Migration

The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton-Watson process with migration introduced in [Yanev & Mitov (1980) C. R. Acad. Bulg. Sci. 33(4):473-475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

math.PR↗
arXiv2024

Diffusion approximation of critical controlled multi-type branching processes

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs.

math.PR↗
arXiv2024

An inhomogeneous controlled branching process

A discrete time branching process where the offspring distribution is generation-dependent, and the number of reproductive individuals is controlled by a random mechanism is considered. This model is a Markov chain but, in general, the transition probabilities are non-stationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: either becomes extinct almost surely or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are provided. Finally rates of growth of the process provided the non-extinction are studied.

math.PR↗
arXiv2024

Branching processes in varying environment with generation-dependent immigration

A branching process in varying environment with generation-dependent immigration is a modification of the standard branching process in which immigration is allowed and the reproduction and immigration laws may vary over the generations. This flexibility makes the process more appropriate to model real populations due to the fact that the stability in the reproductive capacity and in the immigration laws are not usually fulfilled. In this setting, we study the extinction problem, providing a necessary and sufficient condition for the certain extinction of the population. The asymptotic behaviour of the model is analysed for those processes with critical offspring distributions, according with the classification established in Kersting (2017), and when the immigration means stabilize to a positive value. More specifically, we establish that the asymptotic distribution of the process -- under a suitable normalization -- belongs to the gamma distribution family.

math.PR↗
arXiv2023

Hitchin map on even very stable upward flows

We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GL(n) case we classify the type (1,...,1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of quaternionic Grassmannians, 4n-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.

math.AG↗
arXiv2021

Model choice and parameter inference in controlled branching processes

Our purpose is to estimate the posterior distribution of the parameters of interest for controlled branching processes (CBPs) without prior knowledge of the maximum number of offspring that an individual can give birth to and without explicit likelihood calculations. We consider that only the population sizes at each generation and at least the number of progenitors of the last generation are observed, but the number of offspring produced by any individual at any generation is unknown. The proposed approach is two-fold. Firstly, to estimate the maximum progeny per individual we make use of an approximate Bayesian computation (ABC) algorithm for model choice and based on sequential importance sampling with the raw data. Secondly, given such an estimate and taking advantage of the simulated values of the previous stage, we approximate the posterior distribution of the main parameters of a CBP by applying the rejection ABC algorithm with an appropriate summary statistic and a post-processing adjustment. The accuracy of the proposed method is illustrated by means of simulated examples developed with the statistical software R. Moreover, we apply the methodology to two real datasets describing populations with logistic growth. To this end, different population growth models based on CBPs are proposed for the first time.

stat.ME↗