Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramer type moderate deviation theorem for the general self-normalized sum sum_{i=1}^n X_i/(sum_{i=1}^n Y_i^2)^{1/2}, which unifies and extends the classical Cramer (1938) theorem and the self-normalized Cramer type moderate deviation theorems by Jing, Shao and Wang (2003) as well as the further refined version by Wang (2011). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramer type moderate deviation theorems for one-dependent random variables, geometrically beta-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramer type moderate deviation theorems for self-normalized winsorized mean.
Let $(Z_n)_{ngeq0}$ be a supercritical Galton-Watson process. Consider the Lotka-Nagaev estimator for the offspring mean. In this paper, we establish self-normalized Cram{e}r type moderate deviations and Berry-Esseens bounds for the Lotka-Nagaev estimator. The results are believed to be optimal or near optimal.
We view the classical Lindeberg principle in a Markov process setting to establish a universal probability approximation framework by It^{o}s formula and Markov semigroup. As applications, we consider approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by additive Brownian motion, and obtain an approximation error with explicit dependence on the dimension which makes it possible to analyse high dimensional models. We also apply our framework to study stable approximation and normal approximation and obtain their optimal convergence rates (up to a logarithmic correction for normal approximation).
Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erdos-Renyi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cramer-type moderate deviations via Steins method.
A Cramer-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cramer-type moderate result using Steins method when the limiting was a normal distribution. In this paper, Cramer-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Steins coupling. In particular, a Cramer-type moderate deviation theorem is obtained for the general Curie--Weiss model and the imitative monomer-dimer mean-field model.
Steins method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismuts formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Steins exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance.We apply the theorem to the unadjusted Langevin algorithm.
Under the sublinear expectation $mathbb{E}[cdot]:=sup_{thetain Theta} E_theta[cdot]$ for a given set of linear expectations ${E_theta: thetain Theta}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng (2007)s central limit theorem, in a probability space.
Let ${bf R}$ be the Pearson correlation matrix of $m$ normal random variables. The Raos score test for the independence hypothesis $H_0 : {bf R} = {bf I}_m$, where ${bf I}_m$ is the identity matrix of dimension $m$, was first considered by Schott (2005) in the high dimensional setting. In this paper, we study the asymptotic minimax power function of this test, under an asymptotic regime in which both $m$ and the sample size $n$ tend to infinity with the ratio $m/n$ upper bounded by a constant. In particular, our result implies that the Raos score test is rate-optimal for detecting the dependency signal $|{bf R} - {bf I}_m|_F$ of order $sqrt{m/n}$, where $|cdot|_F$ is the matrix Frobenius norm.
Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries from these data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions about the exogeneity of the covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given a certain number of predictors, namely, the distribution of the correlation of a response variable $Y$ with the best $s$ linear combinations of $p$ covariates $mathbf{X}$, even when $mathbf{X}$ and $Y$ are independent. When the covariance matrix of $mathbf{X}$ possesses the restricted eigenvalue property, we derive such distributions for both a finite $s$ and a diverging $s$, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of $mathbf{X}$. Hence, we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where the residuals are from regularized fits. Our approach is then used to construct the upper confidence limit for the maximum spurious correlation and to test the exogeneity of the covariates. The former provides a baseline for guarding against false discoveries and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid. Our techniques and results are illustrated with both numerical examples and real data analysis.
