We consider the continuous time symmetric random walk with a slow bond on Z, which rates are equal to 1/2 for all bonds, except of vertices {−1,0}, associated rate is given by αn−β/2, where α>0 and β∈[0,∞] parameters model. prove here functional central limit theorem bond: if β∈[0,1), then it converges usual Brownian motion. If β∈(1,∞], reflected And at critical value β=1, snapping out motion (...

We derive Edgeworth-type expansions for Poisson stochastic integrals, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which are based on third cumulants instead of the L3 norm term found in the literature. The use of the third cumulant results into a convergence rate faster than the classical Berry-Es...

We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions $$\begin{aligned} \sum _{i\in [n]^d} \theta _i X_i \end{aligned}$$ random tensors $$\varvec{X}=\langle X_i:i\in [n]^d\rangle $$ which are symmetric and exchangeable, whose entries have bounded third moment vanish on diagonal indices. These expressed in terms intrinsic (and easily computable) p...

We consider a random walk Sk with i.i.d. steps on compact group equipped bi-invariant metric. prove quantitative ergodic theorems for the sum ? k=1Nf(Sk) Hölder continuous test functions f, including central limit theorem, law of iterated logarithm and an almost sure approximation by Wiener process, provided that distribution converges to Haar measure in p-Wasserstein metric fast enough. As exa...

This paper analyzes the maximal achievable rate for a given blocklength and error probability over multiple-antenna ambient backscatter channel. The result consists of finite channel coding achievability bound converse legacy system with alphabet constraints multiple-input-multiple-output based on Neyman-Pearson test, Berry-Esseen theorem, Mellin transform. Then, we derive closed-form expressio...

Given a Gromov-hyperbolic group G $G$ endowed with finite symmetric generating set, we study the statistics of counting measures on spheres associated Cayley graph under linear representations . More generally, obtain weak law large numbers for subadditive functions, echoing classical Fekete lemma. For strongly irreducible and proximal representations, prove central limit theorem Berry–Esseen t...

Suppose that X1, . . . ,Xn are independent, identically-distributed random variables of mean zero and variance one. Assume that E|X1| ≤ δ4. We observe that there exist many choices of coefficients θ1, . . . , θn ∈ R with ∑ j θ 2 j = 1 for which sup α,β∈R α<β ∣∣∣∣∣ P  α ≤ n ∑ j=1 θjXj ≤ β   − 1 √ 2π ∫ β α e−t 2/2dt ∣∣∣∣∣ ≤ Cδ 4 n , (1) where C > 0 is a universal constant. Inequality (1) shou...

In many applications, the available data come from a sampling scheme that causes loss of information in terms of left truncation. In some cases, in addition to left truncation, the data are weakly dependent. In this paper we are interested in deriving the asymptotic normality as well as a Berry-Esseen type bound for the kernel density estimator of left truncated and weakly dependent data.

By Louis H.Y. Chen † and Qi-Man Shao ‡ National University of Singapore and University of Oregon §¶ We establish both uniform and non-uniform error bounds of the Berry-Esseen type in normal approximation under local dependence. These results are of an order close to the best possible if not best possible. They are more general or sharper than many existing ones in the literature. The proofs cou...