Berry Esseen Bounds Research Articles

Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

Let $$\mu $$ be a probability measure on $$\textrm{GL}_d(\mathbb {R})$$ and denote by $$S_n:= g_n \cdots g_1$$ the associated random matrix product, where $$g_j$$ ’s are i.i.d.’s with law $$\mu $$ . We study statistical properties of random variables of the form $$\begin{aligned} \sigma (S_n,x) + u(S_n x), \end{aligned}$$ where $$x \in \mathbb {P}^{d-1}$$ , $$\sigma $$ is the norm cocycle and u belongs to a class of admissible functions on $$\mathbb {P}^{d-1}$$ with values in $$\mathbb {R}\cup \{\pm \infty \}$$ . Assuming that $$\mu $$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry–Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $$\mathbb {R}$$ and Hölder continuous target functions on $$\mathbb {P}^{d-1}$$ . As particular cases, we obtain new limit theorems for $$\sigma (S_n,x)$$ and for the coefficients of $$S_n$$ .

Berry–Esseen Bounds of the Quasi Maximum Likelihood Estimators for the Discretely Observed Diffusions

For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, this paper obtains the Berry–Esseen bounds on the rates of convergence to normality of the distributions of the quasi maximum likelihood estimators based on stochastic Taylor approximation, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval, the high-frequency regime. It shows that the higher-order stochastic Taylor approximation-based estimators perform better than the basic Euler approximation in the sense of having smaller asymptotic variance.

Berry-Esseen's Bound for a Bisexual Branching Process in a Random Environment

Berry-Esseen's Bound for a Bisexual Branching Process in a Random Environment

A Berry-Esseen Bound for Nonlinear Statistics with Bounded Differences

In this paper, we obtain an explicit Berry-Esseen bound in the central limit theorem for nonlinear statistics with bounded differences. Some examples are provided as well.

An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds

This paper is concerned with the rate of convergence of the distribution of the sequence {Fn/Gn}, where Fn and Gn are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the estimation problem of parameters occurring in Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs). We develop a new technique to compute the exact rate of convergence on the Kolmogorov distance for the normal approximation of Fn/Gn. As a tool for our work, an Edgeworth expansion for the distribution of Fn/Gn, with an explicitly expressed remainder, will be developed, and this remainder term will be controlled to obtain an optimal bound. As an application, we provide an optimal Berry–Esseen bound of the Maximum Likelihood Estimator (MLE) of an unknown parameter appearing in SDEs and SPDEs.

Non-uniform Berry-Esseen Bounds for Coordinate Symmetric Random Vectors with Applications

This work establishes the nonuniform Berry-Esseen inequality for coordinate symmetric vectors. The nonuniform Lp (p ≥ 1) bound is also established. The main results are applied to projections of random vectors distributed according to a family of measures on the ${\ell _{r}^{n}}$ sphere and the ${\ell _{r}^{n}}$ ball, including cone measure and volume measure.

Berry–Esseen Bounds and Diophantine Approximation

Let S N , N = 1, 2,... be a random walk on the integers, let α be an irrational number and let Z N = {S N α>}, where {·} denotes fractional part. Then Z N , N = 1, 2,... is a random walk on the circle, and from classical results of probability theory it follows that the distribution of Z N converges weakly to the uniform distribution. We determine the precise speed of convergence, which, in addition to the distribution of the elementary step X of the random walk S N , depends sensitively on the rational approximation properties of α.

Berry–Esseen Bounds for Self-Normalized Martingales

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.

Non-uniform Berry–Esseen Bounds for Weighted U-Statistics and Generalized L-Statistics

Weighted U-statistics and generalized L-statistics are commonly used in statistical inference and their asymptotic properties have been well developed. In this paper sharp non-uniform Berry–Esseen bounds for weighted U-statistics and generalized L-statistic are established.

Berry–Esseen Bounds and the Law of the Iterated Logarithm for Estimators of Parameters in an Ornstein–Uhlenbeck Process with Linear Drift

We study the asymptotic behaviors of estimators of the parameters in an Ornstein–Uhlenbeck process with linear drift, such as the law of the iterated logarithm (LIL) and Berry–Esseen bounds. As an application of the Berry–Esseen bounds, the precise rates in the LIL for the estimators are obtained.

Berry–Esseen Bounds and the Law of the Iterated Logarithm for Estimators of Parameters in an Ornstein–Uhlenbeck Process with Linear Drift

We study the asymptotic behaviors of estimators of the parameters in an Ornstein–Uhlenbeck process with linear drift, such as the law of the iterated logarithm (LIL) and Berry–Esseen bounds. As an application of the Berry–Esseen bounds, the precise rates in the LIL for the estimators are obtained.

Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors

For a coordinate symmetric random vector $(Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n$, that is, one satisfying $(Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n)$ for all $(e_1,\ldots,e_n) \in \{-1,1\}^n$, for which $P(Y_i=0)=0$ for all $i=1,2,\ldots,n$, the following Berry Esseen bound to the cumulative standard normal $\Phi$ for the standardized projection $W_\theta=Y_\theta/v_\theta$ of ${\bf Y}$ holds: $$ \sup_{x \in \mathbb{R}}|P(W_\theta \leq x) - \Phi(x)| \leq 2 \sum_{i=1}^n |\theta_i|^3 E| X_i|^3 + 8.4 E(V_\theta^2-1)^2, $$ where $Y_\theta=\theta \cdot {\bf Y}$ is the projection of ${\bf Y}$ in direction $\theta \in \mathbb{R}^n$ with $||\theta||=1$, $v_\theta=\sqrt{\mbox{Var}(Y_\theta)},X_i=|Y_i|/v_\theta$ and $V_\theta=\sum_{i=1}^n \theta_i^2 X_i^2$. As such coordinate symmetry arises in the study of projections of vectors chosen uniformly from the surface of convex bodies which have symmetries with respect to the coordinate planes, the main result is applied to a class of coordinate symmetric vectors which includes cone measure ${\cal C}_p^n$ on the $\ell_p^n$ sphere as a special case, resulting in a bound of order $\sum_{i=1}^n |\theta_i|^3$.

Subtree Sizes in Recursive Trees and Binary Search Trees: Berry–Esseen Bounds and Poisson Approximations

We study the number of subtrees on the fringe of random recursive trees and random binary search trees whose limit law is known to be either normal or Poisson or degenerate depending on the size of the subtree. We introduce a new approach to this problem which helps us to further clarify this phenomenon. More precisely, we derive optimal Berry–Esseen bounds and local limit theorems for the normal range and prove a Poisson approximation result as the subtree size tends to infinity.

Berry–Esseen Bounds for Standardized Subordinators via Moduli of Smoothness

We introduce moduli of smoothness techniques to deal with Berry–Esseen bounds, and illustrate them by considering standardized subordinators with finite variance. Instead of the classical Berry–Esseen smoothing inequality, we give an easy inequality involving the second modulus. Under finite third moment assumptions, such an inequality provides the main term of the approximation with small constants, even asymptotically sharp constants in the lattice case. Under infinite third moment assumptions, we show that the optimal rate of convergence can be simply written in terms of the first modulus of smoothness of an appropriate function, depending on the characteristic random variable of the subordinator. The preceding results are extended to standardized Levy processes with finite variance.

Average case analysis of the Boyer-Moore algorithm

Limit theorems (including a Berry-Esseen bound) are derived for the number of comparisons taken by the Boyer-Moore algorithm for finding the occurrences of a given pattern in a random text. Previou...

A Berry-Esseen Bound for Linear Combinations of Order Statistics

The rate of convergence of the distribution function of a linear combination of order statistics of n independent and identically distributed random variables with a common distribution function F to its normal limit is investigated. Under the assumptions some α 1, , α 2, β 2 > 0 and with some 0 ≤ κ < 4/3 and appropriate moment conditions a Berry-Esseen bound is given. If the coefficients are generated by a sequence of weight functions of a special structure, then the rate is shown to be . Finally, the result is applied for a statistic, which is widely used in auditing.

THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

SummaryIn this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry‐Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n0) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n*) required to have BEB below BEB0, when the observations are dependent. Comparison of n* with n0 reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.