- A Berry-Esseen theorem and Edgeworth expansions for uniformly elliptic inhomogeneous Markov chains(arXiv)
Creator : Dmitry Dolgopyat, Yeor Hafouta
Summary : We show a Berry-Esseen theorem and Edgeworth expansions for partial sums of the shape SN=∑Nn=1fn(Xn,Xn+1), the place {Xn} is a uniformly elliptic inhomogeneous Markov chain and {fn} is a sequence of uniformly bounded capabilities. The Berry-Esseen theorem holds with out further assumptions, whereas expansions of order 1 maintain when {fn} is irreducible, which is an optimum situation. For increased order expansions, we then deal with two conditions. The primary is when the important supremum of fn is of order $O(n^{-be})$ for some $bein(0,1/2)$. On this case it seems that expansions of any order $r<frac1{1–2be}$ maintain, and this situation is perfect. The second case is uniformly elliptic chains on a compact Riemannian manifold. When fn are uniformly Lipschitz steady we present that SN admits expansions of all orders. When fn are uniformly Hölder steady with some exponent $alin(0,1)$, we present that SN admits expansions of all orders $r<frac{1+al}{1-al}$. For Hölder continues capabilities with $al<1$ our outcomes are new additionally for uniformly elliptic homogeneous Markov chains and a single practical f=fn. In actual fact, we present that the situation $r<frac{1+al}{1-al}$ is perfect even within the homogeneous case.
2. Berry-Esseen bounds with targets and Native Restrict Theorems for merchandise of random matrices(arXiv)
Creator : Tien-Cuong Dinh, Lucas Kaufmann, Hao Wu
Summary : Let μ be a likelihood measure on GLd(R) and denote by Sn:=gn⋯g1 the related random matrix product, the place gj’s are i.i.d.’s with regulation μ. We examine statistical properties of random variables of the shape
σ(Sn,x)+u(Snx),
the place x∈Pd−1, σ is the norm cocycle and u belongs to a category of admissible capabilities on Pd−1 with values in R∪{±∞}. Assuming that μ has a finite exponential second and generates a proximal and strongly irreducible semigroup, we receive optimum Berry-Esseen bounds and the Native Restrict Theorem for such variables utilizing a big class of observables on R and Hölder steady goal capabilities on Pd−1. As specific instances, we receive new restrict theorems for σ(Sn,x) and for the coefficients of Sn
