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In this paper, we obtain the Berry-Esseen bound for multivariate normal approximation for the Polyak-Ruppert averaged iterates of the linear stochastic approximation (LSA) algorithm with decreasing step size. Our findings reveal that the fastest rate of normal approximation is achieved when setting the most aggressive step size $\alpha_{k} \asymp k^{-1/2}$. Moreover, we prove the non-asymptotic validity of the confidence intervals for parameter estimation with LSA based on multiplier bootstrap. This procedure updates the LSA estimate together with a set of randomly perturbed LSA estimates upon the arrival of subsequent observations. We illustrate our findings in the setting of temporal difference learning with linear function approximation.

We describe a challenging robotics deployment in a complex ecosystem to monitor a rich plant community. The study site is dominated by dynamic grassland vegetation and is thus visually ambiguous and liable to drastic appearance change over the course of a day and especially through the growing season. This dynamism and complexity in appearance seriously impact the stability of the robotics platform, as localisation is a foundational part of that control loop, and so routes must be carefully taught and retaught until autonomy is robust and repeatable. Our system is demonstrated over a 6-week period monitoring the response of grass species to experimental climate change manipulations. We also discuss the applicability of our pipeline to monitor biodiversity in other complex natural settings., Comment: to be presented at the Workshop on Field Robotics - ICRA 2024

The question of whether the central limit theorem (CLT) holds for the total number of edges in exponential random graph models (ERGMs) in the subcritical region of parameters has remained an open problem. In this paper, we establish the CLT in a subset of the subcritical region known as Dobrushin's uniqueness region. As a result of our proof, we also derive a convergence rate for the CLT and an explicit formula for the asymptotic variance. To establish our main result, we develop Stein's method for the normal approximation for general functionals of nonlinear exponential families of random variables, which is of independent interest. In addition to ERGM, our general theorem can also be applied to other models., Comment: 27 pages

Cram\'er's moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimator used in statistics. In this paper, we establish self-normalized Cram\'{e}r type moderate deviations for martingales under some mile conditions. The result extends an earlier work of Fan, Grama, Liu and Shao [Bernoulli, 2019]. Moreover, applications of our result to Student's statistic, stationary martingale difference sequences and branching processes in a random environment are also discussed. In particular, we establish Cram\'{e}r type moderate deviations for Student's $t$-statistic for branching processes in a random environment., Comment: 24 pages

We establish nonuniform Berry-Esseen (B-E) bounds for Studentized U-statistics of the rate $1/\sqrt{n}$ under a third-moment assumption, which covers the t-statistic that corresponds to a kernel of degree $1$ as a special case. While an interesting data example raised by Novak (2005) can show that the form of the nonuniform bound for standardized U-statistics is actually invalid for their Studentized counterparts, our main results suggest that, the validity of such a bound can be restored by minimally augmenting it with an additive correction term that decays exponentially in $n$. To our best knowledge, this is the first time that valid nonuniform B-E bounds for Studentized U-statistics have appeared in the literature., Comment: Final version accepted to Bernoulli, with some adjustments suitable for arXiv. Compared to the previous version, main changes include: a) A more general "impossibility result" is proved in Section 2. b) Extended discussion about open problems is included in Section 4 after the proof of Theorem 3.1 c) The proof of Theorem 3.2 is moved to Section 5

We take another look at using Stein's method to establish uniform Berry-Esseen bounds for Studentized nonlinear statistics, highlighting variable censoring and an exponential randomized concentration inequality for a sum of censored variables as the essential tools to carry the arguments involved. As an important application, we prove a uniform Berry-Esseen bound for Studentized U-statistics in a form that exhibits the dependence on the degree of the kernel., Comment: This is an improved version, where the B-E bound for the Studentized U-statistics now shows an explicit dependence on the degree of the kernel. Moreover, we added Liqian Zhang as a co-author who helped to work out this improved bound for Studentized U-statistics

Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n \xi_i $ and $ \langle X \rangle_n=\sum_{i=1}^n \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}).$ We prove Cram\'er's moderate deviation expansions for $\displaystyle \mathbf{P}(X_n/\sqrt{\langle X\rangle_n} \geq x)$ and $\displaystyle \mathbf{P}(X_n/\sqrt{ \mathbf{E}X_n^2} \geq x)$ as $n\to\infty.$ Our results extend the classical Cram\'{e}r result to the cases of normalized martingales $X_n/\sqrt{\langle X\rangle_n}$ and standardized martingales $X_n/\sqrt{ \mathbf{E}X_n^2}$, with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed., Comment: 30 pages

Let $X_1,\dots, X_n$ be independent and identically distributed random vectors in $\mathbb{R}^d$. Suppose $\mathbb{E} X_1=0$, $\mathrm{Cov}(X_1)=I_d$, where $I_d$ is the $d\times d$ identity matrix. Suppose further that there exist positive constants $t_0$ and $c_0$ such that $\mathbb{E} e^{t_0|X_1|}\leq c_0<\infty$, where $|\cdot|$ denotes the Euclidean norm. Let $W=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$ and let $Z$ be a $d$-dimensional standard normal random vector. Let $Q$ be a $d\times d$ symmetric positive definite matrix whose largest eigenvalue is 1. We prove that for $0\leq x\leq \varepsilon n^{1/6}$, \begin{equation*} \left| \frac{\mathbb{P}(|Q^{1/2}W|>x)}{\mathbb{P}(|Q^{1/2}Z|>x)}-1 \right|\leq C \left( \frac{1+x^5}{\det{(Q^{1/2})}n}+\frac{x^6}{n}\right) \quad \text{for}\ d\geq 5 \end{equation*} and \begin{equation*} \left| \frac{\mathbb{P}(|Q^{1/2}W|>x)}{\mathbb{P}(|Q^{1/2}Z|>x)}-1 \right|\leq C \left( \frac{1+x^3}{\det{(Q^{1/2})}n^{\frac{d}{d+1}}}+\frac{x^6}{n}\right) \quad \text{for}\ 1\leq d\leq 4, \end{equation*} where $\varepsilon$ and $C$ are positive constants depending only on $d, t_0$, and $c_0$. This is a first extension of Cram\'er-type moderate deviation to the multivariate setting with a faster convergence rate than $1/\sqrt{n}$. The range of $x=o(n^{1/6})$ for the relative error to vanish and the dimension requirement $d\geq 5$ for the $1/n$ rate are both optimal. We prove our result using a new change of measure, a two-term Edgeworth expansion for the changed measure, and cancellation by symmetry for terms of the order $1/\sqrt{n}$., Comment: 33 pages

Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cram\'er 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 Cram\'er (1938) theorem and the self-normalized Cram\'er 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 Cram\'er 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 Cram\'er type moderate deviation theorems for self-normalized winsorized mean.

Let $(Z_n)_{n\geq0}$ 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-Esseen's bounds for the Lotka-Nagaev estimator. The results are believed to be optimal or near optimal.

We establish a Berry--Esseen bound for general multivariate nonlinear statistics by developing a new multivariate-type randomized concentration inequality. The bound is the best possible for many known statistics. As applications, Berry--Esseen bounds for M-estimators and averaged stochastic gradient descent algorithms are obtained., Comment: 54 pages

We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated It\^{o}'s formula and Markov operator. As applications, we study the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler-Maruyama (EM) discretization for multi-dimensional Ornstein-Uhlenbeck stable process, and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance., Comment: Accepted by Annals of Applied Probability

Let $(X _i)_{i\geq1}$ be a stationary sequence. Denote $m=\lfloor n^\alpha \rfloor, 0< \alpha < 1,$ and $ k=\lfloor n/m \rfloor,$ where $\lfloor a \rfloor$ stands for the integer part of $a.$ Set $S_{j}^\circ = \sum_{i=1}^m X_{m(j-1)+i}, 1\leq j \leq k,$ and $ (V_k^\circ)^2 = \sum_{j=1}^k (S_{j}^\circ)^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbb{P}( \sum_{j=1}^k S_{j}^\circ /V_k^\circ \geq x)$ as $n\to \infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed., Comment: 30 pages

Distance correlation has become an increasingly popular tool for detecting the nonlinear dependence between a pair of potentially high-dimensional random vectors. Most existing works have explored its asymptotic distributions under the null hypothesis of independence between the two random vectors when only the sample size or the dimensionality diverges. Yet its asymptotic null distribution for the more realistic setting when both sample size and dimensionality diverge in the full range remains largely underdeveloped. In this paper, we fill such a gap and develop central limit theorems and associated rates of convergence for a rescaled test statistic based on the bias-corrected distance correlation in high dimensions under some mild regularity conditions and the null hypothesis. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality for high-dimensional distance correlation inference in the sense that the accuracy of normal approximation can increase with dimensionality. Moreover, we provide a general theory on the power analysis under the alternative hypothesis of dependence, and further justify the capability of the rescaled distance correlation in capturing the pure nonlinear dependency under moderately high dimensionality for a certain type of alternative hypothesis. The theoretical results and finite-sample performance of the rescaled statistic are illustrated with several simulation examples and a blockchain application., Comment: 67 pages

Resource competition theory predicts coexistence and exclusion patterns based on species R*s, the minimum resource values required for a species to persist. A central assumption of the theory is that all species have equal access to resources. However, many systems are characterized by preemption exploitation, where some species deplete resources before their competitors can access them (e.g., asymmetric light competition, contest competition among animals). We hypothesize that coexistence under preemption requires an R*-preemption tradeoff, i.e., the species with the priority access should have a higher R* (lower efficiency). Thus, we developed an extension of resource competition theory to investigate partial and total preemption (in the latter, the preemptor is unaffected by species with lower preemption rank). We found that an R*-preemption tradeoff is a necessary condition for coexistence in all models. Moreover, under total preemption, the tradeoff alone is sufficient for coexistence. In contrast, under partial preemption, more conditions are needed, which restricts the parameter space of coexistence. Finally, we discussed the implications of our finding for seemingly distinct tradeoffs, which we view as special cases of R*-preemption tradeoff. These tradeoffs include the digger-grazer, the competition-colonization, and tradeoffs related to light competition between trees and understories.

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 Erd\"os-R\'enyi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cram\'er-type moderate deviations via Stein's method., Comment: 35 pages

A Cram\'er-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 Cram\'er-type moderate result using Stein's method when the limiting was a normal distribution. In this paper, Cram\'er-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Stein's coupling. In particular, a Cram\'er-type moderate deviation theorem is obtained for the general Curie--Weiss model and the imitative monomer-dimer mean-field model., Comment: 49 pages

Stein's 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 Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's 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., Comment: We changed the title and corrected some small errors and typos

Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^n\xi_i $ and $[ S]_n=\sum_{i=1}^n \xi_i^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[ S]_n} \geq x)$ as $n\to+\infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.

A Berry-Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry-Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student's statistic and autoregressive process are also discussed.