The Bootstrap for Dynamical Systems

Abstract

Despite their deterministic nature, dynamical systems often exhibit seemingly random behaviour. Consequently, a dynamical system is usually represented by a probabilistic model of which the unknown parameters must be estimated using statistical methods. When measuring the uncertainty of such parameter estimation, the bootstrap stands out as a simple but powerful technique. In this paper, we develop the bootstrap for dynamical systems and establish not only its consistency but also its second-order efficiency via a novel \textit{continuous} Edgeworth expansion for dynamical systems. This is the first time such continuous Edgeworth expansions have been studied. Moreover, we verify the theoretical results about the bootstrap using computer simulations.

arXiv comment51 pages, 4 figures

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• Turing Instability in an Economic-Demographic Dynamical System Can Lead to Pattern Formation on Geographical Scale
• Edgeworth expansions for weakly dependent random variables
• Consistency of support vector machines for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise