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 study the Bahr-Esseen inequality. We show that the Bahr-Esseen inequality holds with exponent p if it holds with exponent [Formula: see text] for the truncated and centered random variables. The Bahr-Esseen inequality is also true if the truncated random variables are acceptable. We then apply the results to obtain weak and strong laws of large numbers and complete convergence.

This paper studies the asymptotic distribution of descents $\des(w)$ in a permutation $w$, and its inverse, distributed according to Mallows measure. The measure is non-uniform probability on permutations introduced study ranked data. Under this measure, are weighted number inversions they contain, with weighting controlled by parameter $q$. main results Berry-Esseen theorem for $\des(w)+\des(w...

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...

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 (...

Abstract We consider 2-dimensional determinantal processes that are rotationinvariant and study the fluctuations of number points in disks. Based on theory mod-phi convergence, we obtain Berry–Esseen as well precise moderate to large deviation estimates for these statistics. These results consistent with Coulomb gas heuristic from physics literature. also functional limit theorems stochastic pr...