In this article, we are interested in the high dimensional normal approximation of $T_n =\Big(\sum_{i=1}^{n}X_{i1}/\Big(\sqrt{\sum_{i=1}^{n}X_{i1}^2}\Big),\dots,$ $\sum_{i=1}^{n}X_{ip}/\Big(\sqrt{\sum_{i=1}^{n}X_{ip}^2}\Big)\Big)$ in $\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\mathcal{A}^{re}=\{\prod_{j=1}^{p}[a_j,b_j]\cap\mathcal{R}:-\infty\leq a_j\leq b_j \leq \infty, j=1,\ldots,p\}$, where $X_1,\dots,X_n$ are non-degenerate independent $p-$dimensional random vectors. We assume that the components of $X_i$ are independent and identically distributed (iid) and investigate the optimal cut-off rate of $\log p$ in the uniform central limit theorem (UCLT) for $T_n$ over $\mathcal{A}^{re}$. The aim is to reduce the exponential moment conditions, generally assumed for exponential growth of the dimension with respect to the sample size in high dimensional CLT, to some polynomial moment conditions. Indeed, we establish that only the existence of some polynomial moment of order $\in [2,4]$ is sufficient for exponential growth of $p$. However the rate of growth of $\log p$ can not further be improved from $o\big(n^{1/2}\big)$ as a power of $n$ even if $X_{ij}$'s are iid across $(i,j)$ and $X_{11}$ is bounded. We also establish near$-n^{-κ/2}$ Berry-Esseen rate for $T_n$ in high dimension under the existence of $(2+κ)$th absolute moments of $X_{ij}$ for $0< κ\leq 1$. When $κ=1$, the obtained Berry-Esseen rate is also shown to be optimal. As an application, we find respective versions for component-wise Student's t-statistic, which may be useful in high dimensional statistical inference.
New nonuniform Berry--Esseen-type bounds for sums of independent random variables are obtained, motivated by recent studies concerning such bounds for nonlinear statistics. The proofs are based on the Chen--Shao concentration techniques within the framework of the Stein method.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry-Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
A Chernoff-type distribution is a nonnormal distribution defined by the slope at zero of the greatest convex minorant of a two-sided Brownian motion with a polynomial drift. While a Chernoff-type distribution is known to appear as the distributional limit in many non-regular statistical estimation problems, the accuracy of Chernoff-type approximations has remained largely unknown. In the present paper, we tackle this problem and derive Berry-Esseen bounds for Chernoff-type limit distributions in the canonical non-regular statistical estimation problem of isotonic (or monotone) regression. The …
Let $H$ be a real separable Hilbert space and $(a_k)_{k\in\mathbb{Z}}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\mathbb{Z}$ by $X_k=\sum_{j\in\mathbb{Z}}a_j(\varepsilon_{k-j})$ where $(\varepsilon_k)_{k\in\mathbb{Z}}$ is a sequence of i.i.d. centered $H$-valued random variables. We investigate the rate of convergence in the CLT for $X$ and in particular we obtain the usual Berry-Esseen's bound provided that $\sum_{j\in\mathbb{Z}}\vert j\vert\|a_j\|_{\mathcal{L}(H)}<+\infty$ and $\varepsilon_0$ belongs to $L_H^{\infty}$.
In this paper, the uniformly asymptotic normality for sample quantiles of associated random variables is investigated under some conditions on the decay of the covariances. We obtain the rate of normal approximation of order $O(n^{-1/2}\log^2 n)$ if the covariances decrease exponentially to $0$. The best rate is shown as $O(n^{-1/3})$ under a polynomial decay of the covariances.
Let $T>0,α>\frac12$. In the present paper we consider the $α$-Brownian bridge defined as $dX_t=-α\frac{X_t}{T-t}dt+dW_t,~ 0\leq t< T$, where $W$ is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter $ α$ based on the continuous observation $\{X_s,0\leq s\leq t\}$ as $t\uparrow T$. We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $\frac{1}{\sqrt{|\log(T-t)|}}$, as $t\uparrow T$. First we compute an upper bound and then find a lower bound with the …
Stratified Permutational Berry--Esseen Bounds and Their Applications to Statistics
Qiuzhen College, Tsinghua University（清华大学邱成桐书院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Department of Statistics, University of California, Berkeley（加州大学伯克利分校统计学系）
Let $X_n=\sum_{i=1}^{\infty}a_iε_{n-i}$, where the $ε_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{-β})$ with $β>1/2$. When $1/2<β<1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,...,x_{d+1})$, $d\ge0$, from ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n,X_{n+1},...,X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n,...,X_{n+d})$, the Berry--Esseen type bo…
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.
New Berry--Esseen-type bounds, with explicit constant factors, for the distribution of the Student statistic and, equivalently, for that of the self-normalized sum of independent zero-mean random variables are obtained. These bounds are compared with the corresponding existing results.
In this work, we provide a $(n/m)^{-1/2}$-rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry--Esseen bounds, which are inspired by the telescoping method of Chen and Shao (2004) and the recursion method of Kuchibhotla and Rinaldo (2020). Performing a dual induction based on the relationships, we obtain tight Berry-E…
In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and the Normal distribution functions are shown to decay at a sub-Gaussian rate.
Let $Z:=\{Z_t,t\geq0\}$ be a stationary Gaussian process. We study two estimators of $\mathbb{E}[Z_0^2]$, namely $\widehat{f}_T(Z):= \frac{1}{T} \int_{0}^{T} Z_{t}^{2}dt$, and $\widetilde{f}_n(Z) :=\frac{1}{n} \sum_{i =1}^{n} Z_{t_{i}}^{2}$, where $ t_{i} = i Δ_{n}$, $ i=0,1,\ldots, n $, $Δ_{n}\rightarrow 0$ and $ T_{n} := n Δ_{n}\rightarrow \infty$. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving $\widehat{f}_T(Z)$ and $\widetilde{f}_n(Z)$. We apply these results to asymptotically stationary Gaussian processes an…
We give a nonuniform Berry-Esseen bound for self-normalized martingales, which bridges the gap between the result of Haeusler (1988) and Fan and Shao (2018). The bound coincides with the nonuniform Berry-Esseen bound of Haeusler and Joos (1988) for standardized martingales. As a consequence, a Berry-Esseen bound is obtained.
A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of asymptotic normality, Berry-Esseen bound, Edgeworth asymptotic expansions under weakened conditions and Cramer type large deviation results are derived.
The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \ge 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that $$ \sup_{z \in \mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left( \frac{1+(\frac{n}{m})^3}{σ_{n,m}} \right) \quad \mbox{for all $n \ge d$ and $m \ge 2$,} $$ where $W_{n,m}$ and $σ_{n,m}^2$ are the standardized count and variance, respectively, of the number of urns with $d$ balls, and $Z$ is a standard normal random variab…
We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form $ξ_N=\frac 1{\sqrt N}\sum_{n=1}^N(F(X(q_1(n)),...,X(q_\ell(n)))-\bar F)$
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.
In this paper, we prove a Berry--Esseen bound with optimal order for self-normalized sums of local dependent random variables under some mild dependence conditions. The proof is based on Stein's method and a randomized concentration inequality. As applications, we obtain optimal Berry--Esseen bounds for $m$-dependent random variables and graph dependency.
In this paper, we establish optimal Berry--Esseen bounds for the generalized $U$-statistics. The proof is based on a new Berry--Esseen theorem for exchangeable pair approach by Stein's method under a general linearity condition setting. As applications, an optimal convergence rate of the normal approximation for subgraph counts in Erdös--Rényi graphs and graphon-random graph is obtained.
We obtain a Berry-Esseen type bound for the distribution of the maximum likelihood estimator of the drift parameter for fractional Ornstein-uhlenbeck type process driven by sub-fractional Brownian motion.
Berry-Esseen-type bounds for total variation and relative entropy distances to the normal law are established for the sums of non-i.i.d. random variables.
For an Ornstein-Uhlenbeck process driven by fractional Brownian motion with Hurst index $H\in [\frac12,\frac34]$, we show the Berry-Esséen bound of the least squares estimator of the drift parameter. We use an approach based on Malliavin calculus given by Kim and Park \cite{kim 3}.
Under correlation-type conditions, we derive upper bounds of order $\frac{1}{\sqrt{n}}$ for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.
Bolthausen used a variation of Stein's method to give an inductive proof of the Berry-Esseen theorem for sums of independent, identically distributed random variables. We modify this technique to prove a Berry-Esseen theorem for character ratios of a random representation of the symmetric group on transpositions. An analogous result is proved for Jack measure on partitions.
We obtain a sharp estimate of the speed of convergence in the Boolean central limit theorem for measures of finite sixth moment. The main tool is a quantitative version of the Stieltjes-Perron inversion formula.
We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati (2012, 2015) and Biermé, Bonami, Nourdin and Peccati (2013), involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs.
We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate $x$'s, and are of exponential decay rate as de la Peña's inequality when $x\rightarrow \infty$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.
We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function $φ$ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein's inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients.
We give estimates on the rate of convergence in the Boolean central limit theorem for the Lévy distance. In the case of measures with bounded support we obtain a sharp estimate by giving a qualitative description of this convergence.
We show, how the classical Berry-Esseen theorem for normal approximation may be used to derive rates of convergence for random sums of centerd, real-valued random variables with respect to a certain class of probability metrics, including the Kolmogorov and the Wasserstein distances. This technique is applied to several examples, including the approximation by a Laplace distribution of a geometric sum of centered random variables with finite third moment, where a concrete Berry-Esseen bound is derived. This bound reduces to a bound of the supposedly optimal order $\sqrt{p}$ in the i.i.d. case.
We give rates of convergence in the Central Limit Theorem for the coefficients and the spectral radius of the left random walk on GLd(R), assuming the existence of an exponential or polynomial moment.
We consider functionals which are weighted averages of the avoidance function of a Poisson process. Using the approach to Stein's method based on Malliavin calculus for Poisson functionals we provide explicit bounds for the Wasserstein distance between these standardized functionals and the standard normal distribution. Our approach relies on closed-form expressions for the action of some Malliavin type operators on avoidance functionals of Poisson processes. As a result we provide Berry-Esseen bounds in the CLT for the volume of the union of balls of a fixed radius around random Poisson cente…
Let $(Z_n)$ be a supercritical branching process in a random environment $ξ= (ξ_n)$. We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for $\log Z_n$ under the annealed law $\mathbb P$. We also improve some earlier results about the harmonic moments of the limit variable $W=lim_{n\to \infty} W_n$, where $W_n =Z_n/ \mathbb{E}_ξ Z_n$ is the normalized population size.
