Related papers: On the nonuniform Berry--Esseen bound
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…
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…
In this note, we prove that there exists a universal constant $c=\frac{43}{50}$ such that for every $k\in \mathbb{N}$ and every $d<k/2$, every $k$-uniform hypergraph on $n$ vertices and with minimum $d$-degree at least…
We obtain non-uniform Berry-Esseen type estimates for several classes of weakly dependent sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding…
The purpose of this paper is to estimate the limiting variance of asymptotically stationary Gaussian processes observed at high frequency, using the second moment estimator (SME). We study rates of convergence of the central limit theorem…
It is shown that \begin{equation*} \mathsf{P}(\|a_1U_1+\dots+a_nU_n\|>u)\le c\,\mathsf{P}(a\|Z_d\|>u) \end{equation*} for all real $u$, where $U_1,\dots,U_n$ are independent random vectors uniformly distributed on the unit sphere in…
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…
Let $\mathbf{v}_i$ be vectors in $\mathbb{R}^d$ and $\{\varepsilon_i\}$ be independent Rademacher random variables. Then the Littlewood-Offord problem entails finding the best upper bound for $\sup_{\mathbf{x} \in \mathbb{R}^d}…
his study presents a novel technique to estimate the computational complexity of sequential decoding using the Berry-Esseen theorem. Unlike the theoretical bounds determined by the conventional central limit theorem argument, which often…
We investigate the convergence rate in the Lyapunov theorem when the third absolute moments exist. By means of convex analysis we obtain the sharp estimate for the distance in the mean metric between a probability distribution and its zero…
We develop techniques for determining an explicit Berry-Esseen bound in the Kolmogorov distance for the normal approximation of a ratio of Gaussian functionals. We provide an upper bound in terms of the third and fourth cumulants, using…
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…
A strong converse bound for constant composition codes of the form $P_{e}^{(n)} \geq 1- A n^{-0.5(1-E_{sc}'(R,W,p))} e^{-n E_{sc}(R,W,p)}$ is established using the Berry-Esseen theorem through the concepts of Augustin information and…
Consider a sum $S_n=v_i\varepsilon_1+\cdots+v_n\varepsilon_{n}$, where $(v_i)^{n}_{i=1}$ are non-zero vectors in $\mathbb{R}^{d}$ and $(\varepsilon_i)^{n}_{i=1}$ are independent Rademacher random variables (i.e.,…
We obtain rates of convergence in limit theorems of partial sums $S_n$ for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the…
Concentration inequalities for the sample mean, like those due to Bernstein, Hoeffding, and Bentkus, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem…
For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i…
Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed (i.i.d.) $d\times d$ real random matrices. For $n\geq 1$ set $G_n = g_n \ldots g_1$. Given any starting point $x=\mathbb R v\in\mathbb{P}^{d-1}$, consider the…
Let $\zeta = \xi + i\xi'$ where $\xi, \xi'$ are iid copies of a mean zero, variance one, subgaussian random variable. Let $N_n$ be a $n \times n$ random matrix with entries that are iid copies of $\zeta$. We prove that there exists a $c \in…
Let the Ornstein-Uhlenbeck process $\{X_t,\,t\geq 0\}$ driven by a fractional Brownian motion $B^H$ described by $d X_t=-\theta X_t dt+ d B_t^H,\, X_0=0$ with known parameter $H\in (0,\frac34)$ be observed at discrete time instants $t_k=kh,…