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Li-Xin Zhang

Publications and source records attributed to Li-Xin Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Limit Laws of the Iterated Logarithm Under Sub-linear Expectations

Let $\{Y_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, and $S_n=\sum_{i=1}^nY_i$. In this paper, we establish a limit law of \begin{align*}\lim_{n\to \infty}\max_{k\le n}\frac{S_k}{\sqrt{2k \log\log n}}. \end{align*} Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of $Y_1,Y_2,\ldots$, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.

math.PR↗
arXiv2026

Theoretical Properties of Covariate-Adaptive Randomization with a Diverging Number of Covariates

Covariate-adaptive randomization procedures are widely used in clinical trials to improve covariate balance. In modern applications, experimenters often have access to many covariates, motivating the need for a theory of covariate-adaptive randomization procedures with a diverging number of covariates. This paper studies two classes of covariate-adaptive randomization procedures, referred to as imbalance-efficient covariate-adaptive randomization and imbalance-robust covariate-adaptive randomization, when the feature dimension diverges. We derive convergence rates for the imbalance of the covariates used in randomization. For both procedures, the imbalance is of a smaller order than that under complete randomization when the feature dimension is $o(n)$, whereas it is of the same order when the feature dimension is $\Omega(n)$. For imbalance-robust covariate-adaptive randomization, we further establish the asymptotic properties of the imbalance of additional covariates and use these results to derive the asymptotic distribution of the difference-in-means estimator for the average treatment effect and construct asymptotically valid confidence intervals. Furthermore, we provide extensive numerical and empirical studies to illustrate the practical relevance of our theoretical results.

stat.ME↗
arXiv2026

Group Efficient Randomized-Adaptive Designs with Delayed and Missing Responses

Response-adaptive randomization designs have attracted much attention in clinical trials. This paper proposes a new class of response-adaptive design which built on the efficient randomized-adaptive design (ERADE) proposed by Hu, Zhang, and He. The original ERADE uses a discrete allocation probability function and leverages the stopping time theory of stochastic processes to establish asymptotic results. It has been proven that the design can reach the Cram\'er-Rao lower bound for any target allocation proportion. This study further expands the original framework by replacing the traditional case-by-case sequential enrollment with group recruitment over fixed time intervals(weekly, biweekly, or monthly), and dynamically updates the allocation probabilities based on the cumulative response information within each group. Meanwhile, to better fit practical application scenarios, we further explicitly consider the situations of randomly missing data and response delay. Theoretical analysis shows that the new design retains all the main asymptotic properties of the original ERADE, and still performs well under the conditions of response delay and missing data. Finally, through simulation studies and the redesign of a real-world clinical trial, the effectiveness and practicality of the proposed method are verified.

math.ST↗
arXiv2026

Chung-type laws of the iterated logarithm for $m$-fold weighted integrated fractional processes

Let $\{B_H(t);t\ge 0\}$ be a fractional Brownian motion of order $H\in (0,1)$, and $J_{m,\alpha}(B_H)$ be the $m$-fold weighted integrals of $B_H$ defined as $$ J_{m,\bm\alpha}(B_H)(t) =\int_0^ts_m^{-\alpha_m}\int_0^{s_m}\cdots s_2^{-\alpha_2}\int_0^{s_2}s_1^{-\alpha_1}B_H(s_1)d s_1\; ds_2\cdots d s_m, $$ where $\alpha_1+\cdots+\alpha_i<H+i$, $i=1,\ldots,m$, $\bm\alpha=\bm\alpha_m=(\alpha_1,\ldots,\alpha_m)$. We show that \begin{align*} \liminf_{T\to \infty} \frac{(\log\log T)^{H+m}}{T^{H+m-\alpha}}\sup_{0\le t\le T}\left|\frac{ J_{m,\bm\alpha}(B_H)(t)}{t^{\alpha-\alpha_1-\cdots-\alpha_m}}\right| = a_H\left( \frac{\kappa_{H+m}}{1-\alpha/(H+m)}\right)^{H+m}\;\; a.s. \end{align*} for all $\alpha<H+m$, and \begin{align*} \liminf_{T\to \infty} & \sqrt{\frac{\log\log\log T}{\log T}} \sup_{1\le t\le T}\left|\int_1^t \frac{J_{m-1, \bm\alpha_{m-1}}(B_H)(s)}{s^{H+m-\alpha_1-\cdots-\alpha_{m-1}}}ds\right| &= \frac{\pi}{2}\frac{\sqrt{\beta(2H,1-H)}}{\prod_{i=1}^{m-1}\big(H+i-\alpha_1-\cdots-\alpha_i\big)}\;\; a.s., \end{align*} where $a_H$ is an explicit constant with $a_{\frac{1}{2}}=1$, $\kappa_{\lambda}$ is a constant which depends only on $\lambda$, and $\beta(a,b)$ is the beta function.In particular, the exact value of a Chung-type law of the iterated logarithm established by Duker, Li and Linde (2000) is found, and as an application, the Chung-type law of the iterated logarithm for the randomized play-the-winner rule is established. The small ball probabilities of \(J_{m, \bm\alpha}(B_H)\) are established to show the liminf behaviors. Similar Chung-type laws of the iterated logarithm and small ball probabilities for a Riemann-Liouville fractional process are also established.

math.PR↗
arXiv2025

On the Law of the Iterated Logarithm for m-dependent stationary random variables under sub-linear expectations

This paper explores the Law of the Iterated Logarithm (LIL) for $m$-dependent sequences under the framework of sub-linear expectations. We first extend existing LIL results to sequences of independent, non-identically distributed random variables under sub-linear expectations. This extension serves as a crucial intermediary step, facilitating the subsequent establishment of the LIL for $m$-dependent stationary sequences. On the other hand, we also establish necessary conditions for $m$-dependent sequences in sub-linear expectation spaces.

math.PR↗
arXiv2025

A Scenario for Origin of Global 4 mHz Oscillations in Solar Corona

We establish a spherically symmetric model of solar atmosphere, which consists of the whole chromosphere and low corona below the $1.25$ solar radius. It is a hydrodynamic model with heating in the chromosphere through an artificial energy flux. We performed a series of simulations with our model and found oscillations with a peak frequency of $\sim$4 $\rm{mHz}$ in the power spectrum. We confirmed that this resulted from the $p$-mode excited in the transition region and amplified in a resonant cavity situated in the height range $\sim$$4\times10^3$--$2\times10^4$ km. This result is consistent with global observations of Alfv\'enic waves in corona and can naturally explain the observational ubiquity of $4\ \rm{mHz}$ without the difficulty of the $p$-mode passing through the acoustic-damping chromosphere. We also confirmed that acoustic shock waves alone cannot heat the corona to the observed temperature, and found mass upflows in the height range $\sim$$7\times10^3$--$7\times10^4$ km in our model, which pumped the dense and cool plasma into the corona and might be the mass supplier for solar prominences.

astro-ph.SR↗
arXiv2024

A stochastic algorithm approach for the elephant random walk with applications

The randomized play-the-winner rule (RPW) is a response-adaptive design proposed by Wei and Durham (1978) for sequentially randomizing patients to treatments in a two-treatment clinical trial so that more patients are assigned to the better treatment as the clinical trial goes on. The elephant random walk (ERW) proposed by Schutz and Trimper (2004) is a non-Markovian discrete-time random walk on $\mathbb Z$ which has a link to a famous saying that elephants can always remember where they have been. The asymptotic behaviors of RPW rule and ERW have been studied in litterateurs independently, and their asymptotic behaviors are very similar. In this paper, we link RPW rule and ERW with the recursive stochastic algorithm. With the help of a recursive stochastic algorithm, we obtain the Gaussian approximation of the ERW and multi-dimensional varying-memory ERW with random step sizes. By the Gaussian approximation, the central limit theorem, precise law of the iterated logarithm, and almost sure central limit theorem are obtained for the multi-dimensional ERW, the multi-dimensional ERW with random step sizes, and their centers of mass for all the diffusive, critical, and superdiffusive regimes. Based on the Gaussian approximation and the small ball probabilities for a new kind of Gaussian process, the precise Chung type laws of the iterated logarithm of the multi-dimensional ERW with random step sizes and its mass of center are also obtained for both the diffusive regime and superdiffusive regime.

math.PR↗
arXiv2024

Strong law of large numbers for $m$-dependent and stationary random variables under sub-linear expectations

The arm of this paper is to establish the strong law of large numbers (SLLN) of $m$-dependent random variables under the framework of sub-linear expectations. We establish the SLLN for a sequence of independent, but not necessarily identically distributed random variables. The study further extends the SLLN to $m$-dependent and stationary sequence of random variables with the condition $C_{\mathbb V}(|X_1|)<\infty$ which is the sufficient and necessary condition of SLLN in the case of independent and identically distributed random variables.

math.PR↗
arXiv2024

Inference for high-dimensional linear expectile regression with de-biased method

In this paper, we address the inference problem in high-dimensional linear expectile regression. We transform the expectile loss into a weighted-least-squares form and apply a de-biased strategy to establish Wald-type tests for multiple constraints within a regularized framework. Simultaneously, we construct an estimator for the pseudo-inverse of the generalized Hessian matrix in high dimension with general amenable regularizers including Lasso and SCAD, and demonstrate its consistency through a new proof technique. We conduct simulation studies and real data applications to demonstrate the efficacy of our proposed test statistic in both homoscedastic and heteroscedastic scenarios.

stat.ME↗
arXiv2023

The limit points of the strong law of large numbers under the sub-linear expectations

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with the finite Choquet expectation, upper mean $\overline{\mu} $ and lower mean $\underline{\mu} $. Then for any Borel-measurable function $\varphi(x_1,\ldots,x_d)$ on $\mathbb R^d$ or continuous function $\varphi(x_1,x_2,\ldots)$ on $\mathbb R^{\mathbb N}$, $\sum_{i=1}^n X_i/n$ converges to $\underline{\mu}\wedge \varphi(X_1,X_2,\ldots)\wedge \overline{\mu}$ with upper capacity $1$. The limits of $\sum_{i=1}^nX_i/n$ can be with upper capacity 1 also a random set with boundaries being continuous functions or finite-dimensional Borel-measurable functions of $(X_1, X_2,\ldots)$.

math.PR↗
arXiv2023

Spiral shocks induced in galactic gaseous disk: hydrodynamic understanding of observational properties of spiral galaxies

We investigate the properties of spiral shocks in a steady, adiabatic, non-axisymmetric, self-gravitating, mass-outflowing accretion disk around a compact object. We obtain the accretion-ejection solutions in a gaseous galactic disk and apply them to the spiral galaxies to investigate the possible physical connections between some galaxy observational quantities. The self-gravitating disk potential is considered following Mestel's (1963) prescription. The spiral shock-induced accretion-ejection solutions are obtained following the point-wise self-similar approach. We observe that the self-gravitating disk profoundly affects the dynamics of the spiral structure of the disk and the properties of the spiral shocks. We find that the observational dispersion between the pitch angle and shear rate and between the pitch angle and star formation rate in spiral galaxies contains some important physical information. There are large differences in star formation rates among galaxies with similar pitch angles, which may be explained by the different star formation efficiencies caused by the distinct galactic ambient conditions.

astro-ph.GA↗
arXiv2023

Central Limit Theorem for m-dependent random variables under sub-linear expectations

M-dependence is a commonly used assumption in the study of dependent sequences. In this paper, central limit theorems for m-dependent random variables under the sub-linear expectations are established based mainly on the conditions of Zhang. They can be regarded as the extension of independent Lindeberg central limit theorem and for proving this, Rosenthal's inequality for m-dependent random variables is obtained. In particular, we extend the results in Li and establish the central limit theorem for m-dependent stationary sequence.

math.PR↗
arXiv2023

Asymptotic Properties of Multi-Treatment Covariate Adaptive Randomization Procedures for Balancing Observed and Unobserved Covariates

Applications of CAR for balancing continuous covariates remain comparatively rare, especially in multi-treatment clinical trials, and the theoretical properties of multi-treatment CAR have remained largely elusive for decades. In this paper, we consider a general framework of CAR procedures for multi-treatment clinal trials which can balance general covariate features, such as quadratic and interaction terms which can be discrete, continuous, and mixing. We show that under widely satisfied conditions the proposed procedures have superior balancing properties; in particular, the convergence rate of imbalance vectors can attain the best rate $O_P(1)$ for discrete covariates, continuous covariates, or combinations of both discrete and continuous covariates, and at the same time, the convergence rate of the imbalance of unobserved covariates is $O_P(\sqrt n)$, where $n$ is the sample size. The general framework unifies many existing methods and related theories, introduces a much broader class of new and useful CAR procedures, and provides new insights and a complete picture of the properties of CAR procedures. The favorable balancing properties lead to the precision of the treatment effect test in the presence of a heteroscedastic linear model with dependent covariate features. As an application, the properties of the test of treatment effect with unobserved covariates are studied under the CAR procedures, and consistent tests are proposed so that the test has an asymptotic precise type I error even if the working model is wrong and covariates are unobserved in the analysis.

math.ST↗
arXiv2022

Convergence of randomized urn models with irreducible and reducible replacement policy

Generalized Friedman urn is one of the simplest and most useful models considered in probability theory. Since Athreya and Ney (1972) showed the almost sure convergence of urn proportions in a randomized urn model with irreducible replacement matrix under the $L\log L$ moment assumption, this assumption has been regarded as the weakest moment assumption, but the necessary has never been shown. In this paper, we study the strong and weak convergence of generalized Friedman urns. It is proved that, when the random replacement matrix is irreducible in probability, the sufficient and necessary moment assumption for the almost sure convergence of the urn proportions is that the expectation of the replacement matrix is finite, which is less stringent than the $L\log L$ moment assumption, and when the replacement is reducible, the $L\log L$ moment assumption is the weakest sufficient condition. The rate of convergence and the strong and weak convergence of non-homogenous generalized Friedman urns are also derived.

math.PR↗
arXiv2022

The moments of the maximum of normalized partial sums related to laws of the iterated logarithm under the sub-linear expectation

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, $S_n=X_1+\ldots+X_n$. We consider the moments of $\max_{n\ge 1}|S_n|/\sqrt{2n\log\log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.

math.PR↗
arXiv2021

On the laws of the iterated logarithm under the sub-linear expectations without the assumption on the continuity of capacities

In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov's converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich and regular enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities. This revision fills a gap in the proof of Proposition 4.2 of arXiv:2103.01390 under an additional assumption.

math.PR↗
arXiv2021

The sufficient and necessary conditions of the strong law of large numbers under the sub-linear expectations

In this paper, by establishing a Borel-Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on $\mathbb R^{\infty}$ under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-liner expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without any assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given. In the version 1, there are errors in the proof of Lemma 2.1 and 2.2. Version 2 corrected the errors under additional conditions, but Corollaries 3.1-3.4 are only shown for the copy of the random variables in a new sub-linear expectation space. In this version, we show that Corollaries 3.1-3.4 remain true for the original random variables and the results for the copy are just special cases.

math.PR↗