Which properties of a random sequence are dynamically sensitive?
type: Item
instance of: scholarly article
title: Which properties of a random sequence are dynamically sensitive?
zbMATH Open document ID: 1021.60055
author: Itai Benjamini
author: Olle Häggström
author: Yuval Peres
author: Jeffrey E. Steif
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: A lot of classical results in probability theory, for instance the strong law of large numbers, the law of iterated logarithm, and so on, concern almost-sure properties of sequences \(\{X_n\}\) of i.i.d. random variables. Consider a sequence of i.i.d. random variables \(\{X_n\}\) with state space \(S\) and common law \(\nu\), where each variable is replaced independently by an independent variable with the same law according to a Poisson clock, i.e., for each \(n\in\mathbb{N},\) let \(X(t):=\{X_n(t)\},\) \(t\geq 0\), be an independent process which at rate 1 replaces its current value by an independent sample from \(\nu.\) The distribution of \(X(t)\) is \(\mu=\nu^{\mathbb{N}}\) for every \(t\geq 0,\) thus any Borel set \(A\subset S^{\mathbb{N}}\) such that \(\mu(A)=1\) satisfies for all \(t\geq 0,\) \(P\{X(t)\in A\}=1\) and \(P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1.\) The almost-sure properties of \(\{X_n\}\), i.e., the subsets \(A\) of \(S^{\mathbb{N}}\) with \(\mu(A)=1\) are classified by the authors as (dynamically) stable or sensitive according to whether or not the property: for all \(t\geq 0,\) \(P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1\) can be strengthened to \(P\{X(t)\in A\text{ for all }t\}=1\). The authors prove that the law of large numbers and the law of iterated logarithm are dynamically stable while run tests are dynamically sensitive. They obtain multi-fractal analysis of exceptional times for run lengths and for prediction. For the simple random walk in \(\mathbb{Z}^d\) the authors prove that the transience is dynamically stable for \(d\geq 5,\) and dynamically sensitive for \(d=3,4.\) Moreover, for \(d=3,4\) the nonempty random set of exceptional times \(t\) where the random walk is recurrent has Hausdorff dimension \((4-d)\slash 2\) a.s.
reviewed by: Roman Urban
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1906113
zbMATH Keywords: dynamical limit theorems
zbMATH Keywords: exceptional times
zbMATH Keywords: run tests
zbMATH Keywords: von Mises-Church randomness
zbMATH Keywords: random walk
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1046294302
OpenAlex ID: W2006630960
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DOI: 10.1214/AOP/1046294302
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description: scientific article; zbMATH DE number 1906113