The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suita...

This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein’s method with the Malliavin calculus of variations on the Poisson space, we derive a bound, which is strictly smaller than what is available in the literature. This is applied to sequences of multiple integral...

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, \omega, L, h)$ are sequences measures $\{\mu_k^z\}_{k=1}^{\infty}$ parametrized by points $z \in M$. For each $z$ in open orbit, we prove central limit theorem for $\mu_k^z$. The center mass $\mu_k^z$ is image under moment map; after re-centering at $0$ and dilating $\sqrt{k}$, re-normalized measure tends centered Gaus...

Let $$\mu $$ be a probability measure on $$\textrm{GL}_d(\mathbb {R})$$ and denote by $$S_n:= g_n \cdots g_1$$ the associated random matrix product, where $$g_j$$ ’s are i.i.d.’s with law . We study statistical properties of variables form $$\begin{aligned} \sigma (S_n,x) + u(S_n x), \end{aligned}$$ $$x \in \mathbb {P}^{d-1}$$ , $$\sigma is norm cocycle u belongs to class admissible functions $...

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 (...

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...

The central limit theorem is considered with respect to the transport distance W2. We discuss an alternative approach to a result of E. Rio, based on a Berry–Esseen-type bound for the entropic distance to the normal distribution. © 2013 Elsevier B.V. All rights reserved. Let X and Z be random variables with distributions F and G, having finite second moments. The Kantorovich distance W2(F ,G) b...

∞ i=1 aiεn−i, where the εi are i.i.d. with mean 0 and at least finite second moment, and the ai are assumed to satisfy |ai| = O(i) with β > 1/2. When 1/2 < β < 1, Xn is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x1, . . . , xd+1), d≥ 0, from R to R, which includes indicator functions and polynomials, the stationary sequence K(Xn,Xn+1, ...

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...