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-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.

Let $\lambda=(\lambda_1, \ldots,\lambda_n)$ be $\beta$-Jacobi ensembles with parameters $p_1, p_2, n$ and $\beta$ while $\beta$ varying with $n.$ Set $\gamma=\lim_{n\to\infty}\frac{n}{p_1}$ and $\sigma=\lim_{n\to\infty}\frac{p_1}{p_2}.$ In this paper, supposing $\lim_{n\to\infty}\frac{\log n}{\beta n}=0,$ we prove that the empirical measures of different scaled $\lambda$ converge weakly to a Wachter distribution, a Marchenko—Pastur law and a semicircle law corresponding to $\sigma\gamma>0, \sigma=0$ or $\gamma=0,$ respectively. We also offer a full large deviation principle with speed $\beta n^2$ and a good rate function to precise the speed of these convergences. As an application, the strong law of large numbers for the extremal eigenvalues of $\beta$-Jacobi ensembles is obtained.

We consider precise deviations for discrete ensembles. For $\beta=2$ case, we first establish an asymptotic formula of the Christoffel-Darboux kernel of the discrete orthogonal polynomials on an infinite regular lattice with weight ${\rm e}^{-NV (x )}$. Then we use the asymptotic formula to get the precise deviations of the extreme value for corresponding ensemble.

We consider a McKean Vlasov backward stochastic differential equation (MVBSDE) of the form $$ Y_t=-F(t,Y_t,Z_t,[Y_t])\,dt+Z_t \,dB_t, \quad Y_T=\xi,$$ where $[Y_t]$ stands for the law of $Y_t$. We show that if $F$ is locally monotone in $y$, locally Lipschitz with respect to $z$ and law's variable, and the monotonicity and Lipschitz constants $\kappa_N, L_N$ are such that $L^2_N+\kappa_N^+=\mathcal{O}(\log(N))$, then the {\rm MVBSDE} has a unique stable solution.

Kink model is developed to analyze the data where the regression function is two-stage piecewise linear with respect to the threshold covariate but continuous at an unknown kink point. In quantile regression for longitudinal data, kink point where the kink effect happens is often assumed to be heterogeneous across different quantiles. However, the kink point tends to be the same across different quantiles, especially in a region of neighboring quantile levels. Incorporating such homogeneity information could increase the estimation efficiency of the common kink point. In this paper, we propose a composite quantile estimation approach for the common kink point by combining information from multiple neighboring quantiles. Asymptotic normality of the proposed estimator is studied. In addition, we also develop a sup-likelihood-ratio test to check the existence of the kink effect at a given quantile level. A test-inversion confidence interval for the common kink point is also developed based on the quantile rank score test. The simulation studies show that the proposed composite kink estimator is more efficient than the single quantile regression estimator. We also illustrate the proposed method via an application to a longitudinal data set on blood pressure and body mass index.

We propose a two-sample test for the mean functions of functional data when the number of bases is much lager than the sample size. The novel test is based on U-statistics which avoids estimating the covariance operator accurately under the high dimensional situation. We further prove the asymptotic normality of our test statistic under both null hypothesis and a local alternative hypothesis. An extensive simulation study is presented which shows that the proposed test works well in comparison with several other methods under the high dimensional situation. An application to egg-laying trajectories of Mediterranean fruit flies data set demonstrates the applicability of the method.

A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get an upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes.

In this paper, we study a class of dynamic games consisting of finite agents under a stochastic growth model with jumps. The jump process in the dynamics of the capital stock of each agent models announcements regarding each agent in the game occur at Poisson distributed random times. The aim of each agent is to maximize her objective functional with mean-field interactions by choosing an optimal consumption strategy. We prove the existence of a fixed point related to the so-called consistence condition as the number of agents goes large. Building upon the fixed point, we establish an optimal feedback consumption strategy for all agents which is in fact an approximating Nash equilibrium which describes strategies for each agent such that no agent has any incentive to change her strategy.

Poincaré inequality has been studied by Bobkov for radial measures, but few are known about the logarithmic Sobolev inequality in the radial case. We try to fill this gap here using different methods:Bobkov's argument and super-Poincaré inequalities, direct approach via $L_1$-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee-Vempala in the log-concave bounded case are refined for radial measures.

This paper focuses on error density estimation in ultrahigh dimensional sparse linear model, where the error term may have a heavy-tailed distribution. First, an improved two-stage refitted crossvalidation method combined with some robust variable screening procedures such as RRCS and variable selection methods such as LAD-SCAD is used to obtain the submodel, and then the residual-based kernel density method is applied to estimate the error density through LAD regression. Under given conditions, the large sample properties of the estimator are also established. Especially, we explicitly give the relationship between the sparsity and the convergence rate of the kernel density estimator. The simulation results show that the proposed error density estimator has a good performance. A real data example is presented to illustrate our methods.

Test of independence between random vectors $X$ and $Y$ is an essential task in statistical inference. One type of testing methods is based on the minimal spanning tree of variables $X$ and $Y$. The main idea is to generate the minimal spanning tree for one random vector $X$, and for each edges in minimal spanning tree, the corresponding rank number can be calculated based on another random vector $Y$. The resulting test statistics are constructed by these rank numbers. However, the existed statistics are not symmetrical tests about the random vectors $X$ and $Y$ such that the power performance from minimal spanning tree of $X$ is not the same as that from minimal spanning tree of $Y$. In addition, the conclusion from minimal spanning tree of $X$ might conflict with that from minimal spanning tree of $Y$. In order to solve these problems, we propose several symmetrical independence tests for $X$ and $Y$. The exact distributions of test statistics are investigated when the sample size is small. Also, we study the asymptotic properties of the statistics. A permutation method is introduced for getting critical values of the statistics. Compared with the existing methods, our proposed methods are more efficient demonstrated by numerical analysis.

This article is concerned with the high-dimensional location testing problem. For highdimensional settings, traditional multivariate-sign-based tests perform poorly or become infeasible since their Type I error rates are far away from nominal levels. Several modifications have been proposed to address this challenging issue and shown to perform well. However, most of modified sign-based tests abandon all the correlation information, and this results in power loss in certain cases. We propose a projection weighted sign test to utilize the correlation information. Under mild conditions, we derive the optimal direction and weights with which the proposed projection test possesses asymptotically and locally best power under alternatives. Benefiting from using the sample-splitting idea for estimating the optimal direction, the proposed test is able to retain type-I error rates pretty well with asymptotic distributions, while it can be also highly competitive in terms of robustness. Its advantage relative to existing methods is demonstrated in numerical simulations and a real data example.

In this paper, we study and analyze the regularized least squares for function-on-function regression model. In our model, both the predictors (input data) and responses (output data) are multivariate functions (with $d$ variables and $\tilde{d}$ variables respectively), and the model coefficient lies in a reproducing kernel Hilbert space (RKHS). We show under mild condition on the reproducing kernel and input data statistics that the convergence rate of excess prediction risk by the regularized least squares is minimax optimal. Numerical examples based on medical image analysis and atmospheric point spread function estimation are considered and tested, and the results demonstrate that the performance of the proposed model is comparable with that of other testing methods.

This paper is a continuation of our recent paper (Electron. J. Probab., 24(141), (2019)) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes (X_t)_t≥0 with branching mechanisms of infinite second moments. In the aforementioned paper, we proved stable central limit theorems for X_t(f) for some functions f of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for X_t(f) for all functions f of polynomial growth. In this note, we show that the limiting stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for X_t(f) for all functions f of polynomial growth.

Let X={X(t) ∈ R^d, t ∈ R^N } be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions. We study the packing dimension of range X(E) under the anisotropic (time variable) metric space (R^N, ρ) and (space variable) metric space (R^d, τ), where E ⊂ R^N is a Borel set. Our results generalize the corresponding results of Estrade, Wu and Xiao (Commun. Stoch. Anal., 5, 41-64 (2011)) for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.

In this paper, we study a new class of equations called mean-field backward stochastic differential equations (BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H > 1/2. First, the existence and uniqueness of this class of BSDEs are obtained. Second, a comparison theorem of the solutions is established. Third, as an application, we connect this class of BSDEs with a nonlocal partial differential equation (PDE, for short), and derive a relationship between the fractional mean-field BSDEs and PDEs.