arXiv subjects

Lê Vǎn Thành

Publications and source records attributed to Lê Vǎn Thành.

11 recordsLinked to original sources

arXiv2025

Non-uniform bounds for non-normal approximation via Stein's method with applications to the Curie--Weiss model and the imitative monomer-dimer model

This paper establishes a non-uniform Berry--Esseen bound for non-normal approximation using Stein's method. The main theorem generalizes the result of the authors in [Comptes Rendus Mathematique, 2024] to the context of non-normal approximation. As an application of the main result, we derive non-uniform Berry--Esseen bounds in non-central limit theorems for the magnetization in the Curie--Weiss model and the imitative monomer-dimer model. These extend some existing results in the literature, including Theorem 2.1 of Chatterjee and Shao [Ann. Appl. Probab., 2011] and Theorem 1 of Chen [J. Math. Physics, 2016].

math.PR↗

arXiv2025

Non-uniform Berry--Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures

This paper establishes a non-uniform Berry--Esseen bound in normal approximation for exchangeable pairs using Stein's method via a concentration inequality approach. The main theorem extends and improves several results in the literature, including those of Eichelsbacher and Löwe [Electron. J. Probab. 15, 2010, 962--988], and Eichelsbacher [arXiv:2404.07587, 2024]. The result is applied to obtain a non-uniform Berry--Esseen bound for the squared-length of the total spin in the mean-field classical $N$-vector models, and a non-uniform Berry--Esseen bound for Jack deformations of the character ratio.

math.PR↗

arXiv2024

On Rio's proof of limit theorems for dependent random fields

This paper presents an exposition of Rio's proof of the strong law of large numbers and extends his method to random fields. In addition to considering the rate of convergence in the Marcinkiewicz--Zygmund strong law of large numbers, we go a step further by establishing (i) the Hsu--Robbins--Erdös--Spitzer--Baum--Katz theorem, (ii) the Feller weak law of large numbers, and (iii) the Pyke--Root theorem on mean convergence for dependent random fields. These results significantly improve several particular cases in the literature. The proof is based on new maximal inequalities that hold for random fields satisfying a very general dependence structure.

math.PR↗

arXiv2022

On weak laws of large numbers for maximal partial sums of pairwise independent random variables

This paper develops Rio's method [C. R. Acad. Sci. Paris Sér. I Math., 1995] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.

math.PR↗

arXiv2022

On a new concept of stochastic domination and the laws of large numbers

Consider a sequence of positive integers $\{k_n,n\ge1\}$, and an array of nonnegative real numbers $\{a_{n,i},1\le i\le k_n,n\ge1\}$ satisfying $\sup_{n\ge 1}\sum_{i=1}^{k_n}a_{n,i}=C_0\in (0,\infty).$ This paper introduces the concept of $\{a_{n,i}\}$-stochastic domination. We develop some techniques concerning this concept and apply them to remove an assumption in a strong law of large numbers of Chandra and Ghosal [Acta. Math. Hungarica, 1996]. As a by-product, a considerable extension of a recent result of Boukhari [J. Theoret. Probab., 2021] is established and proved by a different method. The results on laws of large numbers are new even when the summands are independent. Relationships between the concept of $\{a_{n,i}\}$-stochastic domination and the concept of $\{a_{n,i}\}$-uniform integrability are presented. Two open problems are also discussed.

math.PR↗

arXiv2022

Optimal Bounds in Normal Approximation for Many Interacting Worlds

In this paper, we use Stein's method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert, and Wiseman [Phys. Rev. X. (2014)]. Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin [Ann. Appl. Probab. (2016)].

math.PR↗

arXiv2021

On the complete convergence for sequences of dependent random variables via stochastic domination conditions and regularly varying functions theory

This note develops Rio's proof [C. R. Math. Acad. Sci. Paris, 1995] of the rate of convergence in the Marcinkiewicz--Zygmund strong law of large numbers to the case of sums of dependent random variables with regularly varying normalizing constants. It allows us to obtain a complete convergence result for dependent sequences under uniformly bounded moment conditions. This result is new even when the underlying random variables are independent. The main theorems are applied to three different dependence structures: (i) $m$-pairwise negatively dependent random variables, (ii) $m$-extended negatively dependent random variables, and (iii) $φ$-mixing sequences. To our best knowledge, the results for cases (i) and (ii) are the first results in the literature on complete convergence for sequences of $m$-pairwise negatively dependent random variables and $m$-extended negatively dependent random variables under the optimal moment conditions even when $m=1$. While the results for cases (i) and (iii) unify and improve many existing ones, the result for case (ii) complements the main result of Chen et al. [J. Appl. Probab., 2010]. Affirmative answers to open questions raised by Chen et al. [J. Math. Anal. Appl., 2014] and Wu and Rosalsky [Glas. Mat. Ser. III, 2015] are also given. An example illustrating the sharpness of the main result is presented.

math.PR↗

arXiv2021

On the Baum--Katz theorem for sequences of pairwise independent random variables with regularly varying normalizing constants

This paper proves the Baum--Katz theorem for sequences of pairwise independent identically distributed random variables with general norming constants under optimal moment conditions. The proof exploits some properties of slowly varying functions and the de Bruijn conjugates, and uses the techniques developed by Rio (1995) to avoid using the maximal type inequalities.

math.PR↗

arXiv2021

The Marcinkiewicz--Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences

This paper establishes complete convergence for weighted sums and the Marcinkiewicz--Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables $\{X,X_n,n\ge1\}$ with general normalizing constants under a moment condition that $ER(X)<\infty$, where $R(\cdot)$ is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Statist Probab Lett 92:45--52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijin conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944-1946, 1995) on the Marcinkiewicz--Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrated examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.

math.PR↗

arXiv2020

On the $(p,q)$-type Strong Law of Large Numbers for Sequences of Independent Random Variables

Li, Qi, and Rosalsky (Trans. Amer. Math. Soc., 2016) introduced a refinement of the Marcinkiewicz--Zygmund strong law of large numbers (SLLN), so-called the $(p,q)$-type SLLN, where $0 0$. They obtained sets of necessary and sufficient conditions for this new type SLLN for two cases: $0 p$, and $1\le p<2,q\ge 1$. This paper gives a complete solution to open problems raised by Li, Qi, and Rosalsky by providing the necessary and sufficient conditions for the $(p,q)$-type SLLN for the cases where $0<q\le p<1$ and $0<q<1\le p<2$. We consider random variables taking values in a real separable Banach space $\mathbf{B}$, but the results are new even when $\mathbf{B}$ is the real line. Furthermore, the conditions for a sequence of random variables $\left\{X_n, n \ge 1\right\}$ satisfying the $(p, q)$-type SLLN are shown to provide an exact characterization of stable type $p$ Banach spaces.

math.PR↗

arXiv2019

Error bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models

This paper gives the Kolmogorov and Wasserstein bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models. The Kolmogorov bound is new while the Wasserstein bound improves a result obtained recently by Kirkpatrick and Nawaz [Journal of Statistical Physics, \textbf{165} (2016), no. 6, 1114--1140]. The proof is based on Stein's method for exchangeable pairs.

math.PR↗