Abstract We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation dual and the mean width polyhedral sets are obtained. deduce these results by proving a general theorem weighted volume convex hull points chosen from boundary smooth body according to positive continuous density Euclide...

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.

Edgeworth expansions for random walks on covering graphs with groups of polynomial volume growths are obtained under a few natural assumptions. The coefficients appearing in this expansion depends not only geometric features the underlying but also modified harmonic embedding graph into certain nilpotent Lie group. Moreover, we apply rate convergence Trotter's approximation theorem to establish...

Abstract In this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in classical Hölder inequality for random vectors . We prove central limit theorem ratio, which then allows us to reverse up multiplicative constant with high probability. The models randomness include uniform distribution on balls and spheres. also provide Berry–Esseen–type result larg...

Let (Xt, Yt)t∈T be a discrete or continuous-time Markov process with state space X × R where X is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. (Xt, Yt)t∈T is assumed to be a Markov additive process. In particular, this implies that the first component (Xt)t∈T is also a Markov process. Markov random walks or additive f...

This paper proves a Berry–Esseen theorem for sample quantiles of strongly-mixing random variables under a polynomial mixing rate. The rate of normal approximation is shown to be O(n) as n→∞, where n denotes the sample size. This result is in sharp contrast to the case of the sample mean of strongly-mixing random variables where the rate O(n) is not known even under an exponential strong mixing ...

In an attempt to investigate the adequacy of the normal approximation for the number of nuclei in certain growth/coverage models, we consider a Markov chain which has properties in common with related continuous-time Markov processes (as well as being of interest in its own right). We establish that the rate of convergence to normality for the number of \drops" during times 1; 2; : : : ; n is o...

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 suucient to distinguish between saturated and non-saturated regimes for the network. Several application...

A binary classification problem is considered, where the posteriori probability is estimated by the nonparametric kernel regression estimate with naive kernel. The excess error probability of the corresponding plug-in decision classification rule according to the error probability of the Bayes decision is studied such that the excess error probability is decomposed into approximation and estima...