arXiv subjects

Carlos Fernández

Publications and source records attributed to Carlos Fernández.

6 recordsLinked to original sources

arXiv2026

Christoffel words as extremal structures in Collatz dynamics

We study the combinatorial structure of parity sequences associated with the accelerated Collatz map with the goal of identifying extremal configurations and relating them to the existence of periodic orbits. To each finite sequence of an orbit, we associate a binary word whose ones encode the odd iterates, and we introduce a functional $C(d)$ on such words which provides an explicit expression for the iterates and characterizes possible periodic cycles. We define a natural rotation action on binary words, compatible with the cyclic structure of periodic orbits, and consider the functional $C_{\min}(d)$ as a canonical representative of each rotation class. In this setting, we formulate and solve a discrete optimization problem on the set of binary words of fixed length and prescribed density. We prove that Christoffel words are, up to rotation, the unique maximizers of $C_{\min}(d)$ on $D_{N,r}$, the set of binary words of length $N$ with exactly $r$ ones, thereby establishing a direct connection between the dynamics of the Collatz problem and the classical theory of balanced words. As a consequence, we obtain restrictions on the possible existence of nontrivial cycles and derive explicit bounds for the minimum element of an orbit in terms of its length and the proportion of odd iterates. These results show that the combinatorial structure of parity sequences imposes strong constraints on Collatz dynamics and suggest that extremal configurations are governed by classical objects from the combinatorics on words, exhibiting a pronounced structural rigidity.

math.DS↗

arXiv2025

Anomaly Detection based on Markov Data: A Statistical Depth Approach

The purpose of this article is to extend the notion of statistical depth to the case of sample paths of a Markov chain. Initially introduced to define a center-outward ordering of points in the support of a multivariate distribution, depth functions permit to generalize the notions of quantiles and (signed) ranks for observations in $\mathbb{R}^d$ with $d>1$, as well as statistical procedures based on such quantities. Here we develop a general theoretical framework for evaluating the depth of a Markov sample path and recovering it statistically from an estimate of its transition probability with (non-) asymptotic guarantees. We also detail some of its applications, focusing particularly on unsupervised anomaly detection. Beyond the theoretical analysis carried out, numerical experiments are displayed, providing empirical evidence of the relevance of the novel concept we introduce here to quantify the degree of abnormality of Markov paths of variable length.

math.ST↗

arXiv2025

Tail Index Estimation for Discrete Heavy-Tailed Distributions

It is the purpose of this paper to investigate the issue of estimating the regularity index $β>0$ of a discrete heavy-tailed r.v. $S$, \textit{i.e.} a r.v. $S$ valued in $\mathbb{N}^*$ such that $\mathbb{P}(S>n)=L(n)\cdot n^{-β}$ for all $n\geq 1$, where $L:\mathbb{R}^*_+\to \mathbb{R}_+$ is a slowly varying function. As a first go, we consider the situation where inference is based on independent copies $S_1,\; \ldots,\; S_n$ of the generic variable $S$. Just like the popular Hill estimator in the continuous heavy-tail situation, the estimator $\widehatβ$ we propose can be derived by means of a suitable reformulation of the regularly varying condition, replacing $S$'s survivor function by its empirical counterpart. Under mild assumptions, a non-asymptotic bound for the deviation between $\widehatβ$ and $β$ is established, as well as limit results (consistency and asymptotic normality). Beyond the i.i.d. case, the inference method proposed is extended to the estimation of the regularity index of a regenerative $β$-null recurrent Markov chain. Since the parameter $β$ can be then viewed as the tail index of the (regularly varying) distribution of the return time of the chain $X$ to any (pseudo-) regenerative set, in this case, the estimator is constructed from the successive regeneration times. Because the durations between consecutive regeneration times are asymptotically independent, we can prove that the consistency of the estimator promoted is preserved. In addition to the theoretical analysis carried out, simulation results provide empirical evidence of the relevance of the inference technique proposed.

math.ST↗

arXiv2024

Harris recurrent Markov chains and nonlinear monotone cointegrated models

In this paper, we study a nonlinear cointegration-type model of the form $Z_t = f_0(X_t) + W_t$ where $f_0$ is a monotone function and $X_t$ is a Harris recurrent Markov chain. We use a nonparametric Least Square Estimator to locally estimate $f_0$, and under mild conditions, we show its strong consistency and obtain its rate of convergence. New results (of the Glivenko-Cantelli type) for localized null recurrent Markov chains are also proved.

math.ST↗

arXiv2024

Regenerative bootstrap for $β$-null recurrent Markov chains

Two regeneration-based bootstrap methods, namely, the \textit{Regeneration based-bootstrap} \cite{AthreyaFuh1992, Somnat-1993} and the \textit{Regenerative Block bootstrap} \cite{Bertail2006} are shown to be valid for the problem of estimating the integral of a function with respect to the invariant measure in a $β$-null recurrent Markov chain with an accessible atom. An extension of the Central Limit Theorem for randomly indexed sequences is also presented.

math.ST↗

arXiv2022

Quantification of internal dosimetry in PET patients II: Individualized Monte Carlo-based dosimetry for [18F]Fluorocholine PET

Purpose: To obtain individualized internal doses with a Monte Carlo method in patients undergoing diagnostic FCH-PET studies and to compare such doses with the MIRD method calculations. Methods: A patient cohort of 17 males were imaged after intravenous administration of a mean FCH activity of 244.3 MBq. The resulting PET/CT images were processed in order to generate individualized input source and geometry files for dose computation with the MC tool GATE. The resulting dose estimates were studied and compared to the MIRD method with two different computational phantoms. Mass correction of the S-factors was applied when possible. Potential sources of uncertainty were closely examined: the effect of partial body images, urinary bladder emptying, and biokinetic modeling. Results: Large differences in doses between our methodology and the MIRD method were found, generally in the range of $\pm$25%, and up to $\pm$120% for some cases. The mass scaling showed improvements, especially for non-walled and high-uptake tissues. Simulations of the urinary bladder emptying showed negligible effects on doses to other organs, with the exception of the prostate. Dosimetry based on partial PET/CT images (excluding the legs) resulted in an over-estimation of mean doses to bone, skin, and remaining tissues, and minor differences in other organs/tissues. Estimated uncertainties associated with the biokinetics of FCH introduce variations of cumulated activities in the range of $\pm$10% in the high-uptake organs. Conclusions: The MC methodology allows for a higher degree of dosimetry individualization than the MIRD methodology, which in some cases leads to important differences in dose values. Dosimetry of FCH-PET based on a single partial PET study seems viable due to the particular biokinetics of FCH, even though some correction factors may need to be applied to estimate mean skin/bone doses.

physics.med-ph↗