Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$ in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen (Ann. Probab. 1982) and Grama and Haeusler (Stochastic Process. Appl. 2000), is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.
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 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.
We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form $ξ_N=\frac 1{\sqrt N}\sum_{n=1}^N(F(X(q_1(n)),...,X(q_\ell(n)))-\bar F)$
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.
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.
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.
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 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.
For an Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst parameter 0<H<1/2, one shows the Berry-Esséen bound of the least squares estimator of the drift parameter. Thus, a problem left in the previous paper (Chen, Kuang and Li in Stochastics and Dynamics, 2019+) is solved, where the Berry-Esséen bound of the least squares estimator is proved for 1/2<=H<=3/4. An approach based on Malliavin calculus given by Kim and Park \cite{kim 3} is used
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.
We obtain Berry-Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order $δ\in (2,\infty]$ using a Fourier transform approach. Our bounds improve the state-of-the-art in the regime where the degree of the dependency graph is large. As a Corollary of our results, we obtain a Central Limit Theorem for random variables with a sparse dependency graph that are uniformly bounded in $L^δ$ for some $δ\in(2,\infty]$.
Berry-Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin-Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small-ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.
\noindent We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. We prove a Berry-Esseen bound in a general setting and a large deviation result when the Laplace transform of the underlying distribution is not defined in a neighborhood of zero. Then we present several combinatorial applications. In particular, we prove a large deviation result for the model of hashing with linear probing.
Applying Stein's method, an inductive technique and size bias coupling yields a Berry-Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdos-Renyi random graph of a given degree.
Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metr…
Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices and consider the matrix product $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and moderate deviation expansions of Cramér type, for the matrix norm $\| G_n \|$ of $G_n$, for its $(i,j)$-th entry $G_n^{i,j}$, and the and for its spectral radius $ρ(G_n)$.
Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of $n^{-1/2}$ in the uniform metric, in other words, when do we have the optimal Berry-Esseen bound? We study this question in a quite general framework and find the (almost) sharp dependence conditions. The result applies to many different processes and dynamical systems. As specific, prominent examples, we study functions of the doubling map 2x mod 1 and the left random walk on the general linear group.
We study the inhomogeneous Curie-Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.
We analyze the quality of the gaussian approximation to linear combinations of n independent, identically-distributed random variables with finite fourth moments. It turns out that there exist universal, simple linear combinations that perform better than the sum of the variables. We also investigate the case in which the random variables are independent, yet they are not necessarily identically distributed.
We give a simple technic to derive the Berry-Esséen bounds for the quadratic variation of the subfractional Brownian motion (subfBm). Our approach has two main ingredients: ($i$) bounding from above the covariance of quadratic variation of subfBm by the covariance of the quadratic variation of fractional Brownian motion (fBm); and ($ii$) using the existing results on fBm in \cite{BN08,NP09,N12}. As a result, we obtain simple and direct proof to derive the rate of convergence of quadratic variation of subfBm. In addition, we also improve this rate of convergence to meet the one of fractional Br…
Certain smoothing inequalities were proposed in the recent paper posted on arXiv at arxiv:1301.2828 in order to lessen the very large gap between the best correctly established upper and lower bounds on the constant factor in the nonuniform Berry-Esseen bound. As an illustration of the possible uses of those inequalities, a quick proof of Nagaev's classical nonuniform bound was given there. Here we describe another, apparently more effective class of smoothing inequalities, and give a yet quicker and simpler proof of Nagaev's result.
Berry Esseen type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds are applied to the occurrences of fixed relatively ordered sub-sequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs on finite …
It is shown that the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\le n\le 500000$, where $n$ is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009--2011, an upper bound is obtained for the absolute constant in the Berry--Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.0…
We prove Berry-Esseen type rates of convergence for central limit theorems (CLTs) of regenerative processes which generalize previous results of Bolthausen under weaker moment assumptions. We then show how this general result can be applied to obtain rates of convergence for (1) CLTs for additive functionals of positive recurrent Markov chains under certain conditions on the strong mixing coefficients, and (2) annealed CLTs for certain ballistic random walks in random environments.
We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on chaos expansions methods. Our results apply to $U$-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate $U$-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weigh…
We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein's method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.
Let $A_n= \varepsilon_n \cdots \varepsilon_1$, where $(\varepsilon_n)_{n \geq 1}$ is a sequence of independent random matrices taking values in $ GL_d(\mathbb R)$, $d \geq 2$, with common distribution $μ$. In this paper, under standard assumptions on $μ$ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for $\log ( \Vert A_n \Vert)$ when $μ$ has a polynomial moment. More precisely, we get the rate $((\log n) / n)^{q/2-1}$ when $μ$ has a moment of order $q \in ]2,3]$ and the rate $1/ \sqrt{n} $ when $μ$ has a moment of order $4$, which significantly improves earlier r…
We consider a branching random walk on $d$-dimensional real space with immigration in a time-dependent random environment. Let $Z_n(\mathbf t)$ be the so-called partition function of the process, namely, the moment generating function of the counting measure describing the dispersion of individuals at time $n$. For $\mathbf t$ fixed, the logarithm $\log Z_n(\mathbf t)$ satisfies a central limit theorem. By studying the logarithmic moments of the intrinsic submartingale of the system and its convergence rates, we establish the uniform and non-uniform Berry-Esseen bounds corresponding to the cen…
By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$ρ(F_n,Φ)\le\frac{0.335789(β^3+0.425)}{\sqrt{n}}$$ and $$ρ(F_n,Φ)\le \frac{0.3051(β^3+1)}{\sqrt{n}} $$ are proved for the uniform distance $ρ(F_n,Φ)$ between the standard normal distribution function $Φ$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $β^3$. The first of these inequalities sharpens the best known version of the clas…
This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations on the Poisson space, we derive a bound, which is strictly smaller than what is available in the literature. This is applied to sequences of multiple integrals and sequences of Poisson functionals having a finite chaotic expansion. This leads to new Berry-Esseen bounds in de Jong's theorem for degenerate U-statistics. Moreover, geometric functionals of inters…
In this paper we investigate the speed of convergence of the fluctuations of a general class of Feynman-Kac particle approximation models. We design an original approach based on new Berry-Esseen type estimates for abstract martingale sequences combined with original exponential concentration estimates of interacting processes. These results extend the corresponding statements in the classical theory and apply to a class of branching and genealogical path-particle models arising in nonlinear filtering literature as well as in statistical physics and biology.
Let $B$ be a bifractional Brownian motion with parameters $H\in (0, 1)$ and $K\in(0,1]$. For any $n\geq1$, set $Z_n =\sum_{i=0}^{n-1}\big[n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\E((B_{i+1}-B_{i})^2)\big]$. We use the Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati \cite{NP09} to derive, in the case when $0<HK\leq3/4$, Berry-Esséen-type bounds for the Kolmogorov distance between the law of the correct renormalization $V_n$ of $Z_n$ and the standard normal law. Finally, we study almost sure central limit theorems for the sequence $V_n$.
Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\geq 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise variance expansion for the vertex number is determined up to the constant-order term, a result which can be considered as a second-order analogue of the classical expansion for the expectation of Rényi and Sulanke (1963). Moreover, a sharp Berry-Esseen bound is derived for the vertex number of the random polygon $P_n$, which is of the same order as the square-…
In this paper, we study the asymptotic behavior of the quadratic variation for the class of AR(1) processes driven by white noise in the second Wiener chaos. Using tools from the analysis on Wiener space, we give an upper bound for the total-variation speed of convergence to the normal law, which we apply to study the estimation of the model's mean-reversion. Simulations are performed to illustrate the theoretical results.
This note provides a conditional Berry-Esseen bound for the sum of a martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to a filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate the conditional distribution of $S=\sum_{i=1}^n X_i$ given some $σ$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that of a mean-zero normal random vector having the same conditional variance given $\mathcal{F}_0 $ as the vector $S$. Assuming that the conditional variances $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge 1$, are $\mathcal{F}_0$-measurable and non-singular, an…
Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under the annealed law, uniformly in $n_0 \in \mathbb{N}$, which extend the corresponding results by Grama et al. (Stochastic Process.\ Appl. 2017) established for $n_0=0$. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. (2017) are relaxed in our present setting. An example of appl…
Consider the sum $Z = \sum_{n=1}^\infty λ_n (η_n - \mathbb{E}η_n)$, where $η_n$ are i.i.d.~gamma random variables with shape parameter $r > 0$, and the $λ_n$'s are predetermined weights. We study the asymptotic behavior of the tail $\sum_{n=M}^\infty λ_n (η_n - \mathbb{E}η_n)$ which is asymptotically normal under certain conditions. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. We illustrate the effectiveness of these expansions on an infinite sum of weighted chi-squared distributions.
