Random f-expansions
type: Item
instance of: scholarly article
title: Random f-expansions
zbMATH Open document ID: 0658.60050
author: Jon Aaronson
published in: The Annals of Probability
publication date: 1986-01-01T00:00:00Z
review text: A stationary process \(\{\phi_ n\}^{\infty}_{n=0}\) defined on a probability space (\(\Omega\),A,P) is said to be continued fraction mixing if there exists a sequence \(\{\epsilon_ n\}\) of nonnegative real numbers with \(\lim_{n\to \infty}\epsilon_ n=0\) such that, for all \(m,n\in N\), \(A\in \sigma (\{\phi_ k:k\leq m\})\) and \(B\in \sigma (\{\phi_ k:k\geq m+n\}),\) \[ | P(A\cap B)-P(A)\cdot P(B)| \leq \epsilon_ n\cdot P(A)\cdot P(B). \] The following generalization of a theorem of P. Lévy on continued fraction expansions is proved. Suppose that \(\{\phi_ n\}^{\infty}_{n=0}\) is a nonnegative, continued fraction mixing stationary process defined on (\(\Omega\),A,P) such that \(E(\phi_ 0)=\infty\), and that \(L(X)=E(\phi_ 0\wedge x)\) is regularly varying with index 1-\(\gamma\) as \(x\uparrow \infty\) \((0<\gamma \leq 1)\). Then \[ dist (b(n))^{-1}\sum^{n}_{k=1}\phi_ k\to dist Y_{\gamma}\quad as\quad n\to \infty, \] where b(n)\(\sim n\cdot L(b(n))\) and \(Y_{\gamma}\) is a nonnegative random variable with Laplace transform \(E(e^{-pY_{\gamma}})=e^{-\Gamma (2-\gamma)p^{\gamma}}\) \((p>0)\). The theorem is proved by showing that the base of the tower over the associated shift transformation with height function \([\phi_ 0]\) is a Darling-Kac set, defined as follows. Let (X,\({\mathcal B},m,T)\) be a conservative ergodic measure-preserving transformation on a \(\sigma\)-finite nonatomic measure space and let \(\hat T\) be the operator on \(L^ 1(X,B,m)\) dual to T. Then a set \(A\in B\), \(0<m(A)<\infty\), is called a Darling-Kac (DK) set for T if there are constants \(a_ n>0\) such that \[ \| a_ n^{- 1}\sum^{n}_{k=1}\hat T^ k1_ A-1\|_{L^{\infty}(A)}\to 0\quad as\quad n\to \infty. \] For DK sets an asymptotic renewal equation and a distributional limit theorem for the number of visits (due to D. A. Darling and M. Kac) hold. These are the basic properties of DK sets used in the proof of the theorem. The main lemma of the paper gives sufficient conditions for a set to …
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 4076192
zbMATH Keywords: continued fraction expansions
zbMATH Keywords: measure-preserving transformation
zbMATH Keywords: asymptotic renewal equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992457
OpenAlex ID: W1993351886
DOI: 10.1214/AOP/1176992457
description: scientific article; zbMATH DE number 4076192