arXiv subjects

Santiago Arenas-Velilla

Publications and source records attributed to Santiago Arenas-Velilla.

7 recordsLinked to original sources

arXiv2026

Distributional Convergence of Empirical Entropic Optimal Transport and Statistical Applications

Recently, the statistical properties of empirical Entropic Optimal Transport (EOT) have attracted great interest, as this quantity has been shown to be useful for complex data analysis, among other reasons due to its computational efficiency. In several applications, it has been observed that the EOT plan provides valuable information beyond just the optimal value. For example, in cell biology, colocalization analysis based on the EOT plan has been introduced as a measure for quantification of spatial proximity of different protein assemblies. Despite recent progress in the analysis of its risk properties, a precise understanding of its statistical fluctuations to make it accessible for inference remains elusive to a large extent. In this paper, we derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method. As an application, we obtain uniform confidence bands for colocalization curves and bootstrap consistency. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.

math.ST↗

arXiv2024

Central limit theorem for crossings in randomly embedded graphs

We consider the number of crossings in a random embedding of a graph, $G$, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of $G$. Using Stein's method and size-bias coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where $G$ is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random $d$-regular graphs, and mixtures of random graphs.

math.PR↗

arXiv2024

Extreme eigenvalues of Laplacian random matrices with Gaussian entries

A Laplacian matrix is a real symmetric matrix whose row and column sums are zero. We investigate the limiting distribution of the largest eigenvalues of a Laplacian random matrix with Gaussian entries. Unlike many classical matrix ensembles, this random matrix model contains dependent entries. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics. In particular, after properly shifting and scaling, we show that the largest eigenvalue converges to the Gumbel distribution as the dimension of the matrix tends to infinity. While the largest diagonal entry is also shown to have Gumbel fluctuations, there is a rather surprising difference between its deterministic centering term and the centering term required for the largest eigenvalues.

math.PR↗

arXiv2022

On the links between Stein transforms and concentration inequalities for dependent random variables

In this paper, we explore some links between transforms derived by Stein's method and concentration inequalities. In particular, we show that the stochastic domination of the zero bias transform of a random variable is equivalent to sub-Gaussian concentration. For this purpose a new stochastic order is considered. In a second time, we study the case of functions of slightly dependent light-tailed random variables. We are able to recover the famous McDiarmid type of concentration inequality for functions with the bounded difference property. Additionally, we obtain new concentration bounds when we authorize a light dependence between the random variables. Finally, we give a analogous result for another type of Stein's transform, the so-called size bias transform.

math.PR↗

arXiv2022

Convergence Rate for The Number of Crossing in a Random Labelled Tree

We consider the number of crossings in a random labelled tree with vertices in convex position. We give a new proof of the fact that this quantity is asymptotically Gaussian with mean $n^2/6$ and variance $n^3/45$. Furthermore, we give an estimate for the Kolmogorov distance to a Gaussian distribution which implies a convergence rate of order $n^{-1/2}$.

math.PR↗

arXiv2022

Crossings in Randomly Embedded Graphs

We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various families of graphs, including random chord diagrams or full cycles.

math.CO↗

arXiv2021

Extremal Laws for Laplacian Random Matrices

For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is proven that the fluctuations of the largest eigenvalue and the largest diagonal entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non-asymptotic estimates and bounds for the largest eigenvalue of $L$ in terms of the largest diagonal element of $L$. An expository review of existing results for the asymptotic spectrum of a Laplacian random matrix is also presented, with the goal of noting the differences from the corresponding classical results for Wigner random matrices. Extensions to Laplacian block random matrices are indicated.

math.PR↗