Related papers: Central limit theorems for bounded random variable…
Peng (2006) initiated a new kind of central limit theorem under sub-linear expectations. Song (2017) gave an estimate of the rate of convergence of Peng's central limit theorem. Based on these results, we establish a new kind of almost sure…
In this note, we give a probabilistic interpretation of the Central Limit Theorem used for approximating isotropic Gaussians in [1].
We consider a finite sequence of random points in a finite domain of a finite-dimensional Euclidean space. The points are sequentially allocated in the domain according to a model of cooperative sequential adsorption. The main peculiarity…
We study central limit theorems for certain nonlinear sequences of random variables. In particular, we prove the central limit theorems for the bounded conductivity of the random resistor networks on hierarchical lattices.
A strengthened version of the central limit theorem for discrete random variables is established, relying only on information-theoretic tools and elementary arguments. It is shown that the relative entropy between the standardised sum of…
The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of…
Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian…
We establish central limit theorems for the position and velocity of the charged particle in the mechanical particle model introduced in the paper "Limit velocity for a driven particle in a random medium with mass aggregation"…
The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes. The central limit theorem and functional central limit theorem are obtained for martingale like random variables under…
The Central Limit Theorem states that, in the limit of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to a stable distribution. The…
We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that…
The standard central limit theorem with a Gaussian attractor for the sum of independent random variables may lose its validity in presence of strong correlations between the added random contributions. Here, we study this problem for…
We obtain the analogue of the classical result by Erd\"os and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central…
A central limit theorem is proved for some strictly stationary sequences of random variables that satisfy certain mixing conditions and are subjected to the "shrinking operators" $U_r(x):=[\max\{|x|-r,0\}]\cdot x/|x|,\ r \ge 0$. For…
In Part I of this article (Banerjee and Kuchibhotla (2023)), we have introduced a new method to bound the difference in expectations of an average of independent random vector and the limiting Gaussian random vector using level sets. In the…
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.
Under the sublinear expectation $\mathbb{E}[\cdot]:=\sup_{\theta\in \Theta} E_\theta[\cdot]$ for a given set of linear expectations $\{E_\theta: \theta\in \Theta\}$, we establish a new law of large numbers and a new central limit theorem…
A non-classical formulation of the central limit theorem is given for sequences of independent random variables with finite second moments. Singular sequences whose members all have a degenerate or normal distribution are excluded from…
H\"ormann (2006) gave an extension of almost sure central limit theorem for bounded Lipschitz 1 function. In this paper, we show that his result of almost sure central limit theorem is also hold for any Lipschitz function under stronger…
Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the symmetric group. We generalize this result in several ways. We…