This note provides a conditional Berry-Esseen bound for the sum of martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate distribution $S=\sum_{i=1}^n X_i$ given some $\sigma$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that mean-zero normal random vector having same variance $\mathcal{F}_0 $ as $S$. Assu...

Suppose that the (normalised) partial sum of a stationary sequence converges to standard normal random variable. Given sufficiently moments, when do we have rate convergence n−1∕2 in uniform metric, other words, optimal Berry-Esseen bound? We study this question quite general framework and find (almost) sharp dependence conditions. The result applies many different processes dynamical systems. ...

We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on a continuous-time integration by parts setting and discrete chaos expansions methods. Our approach improves related results obtained in discrete-time settings applies to U-statistics satisfying the weak assumption decomposability Hoeffding sense, yield Kolmogorov distance instead Wasser...

The asymptotic behaviour of a closed BCMP network, with n queues and m n clients, is analyzed when n and m n become simultaneously large. Our method relies on Berry-Esseen type approximations coming in the Central Limit Theorem. We construct critical sequences m 0 n , which are necessary and sufficient to distinguish between saturated and non-saturated regimes for the network. Several applicati...

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

Abstract Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds estimator. The results are believed to optimal or near-optimal.