Limit theorems and ergodicity for general bootstrap random walks*

Andrea Collevecchio, Kais Hamza, Meng Shi, Ruth J. Williams

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Given the increments of a simple symmetric random walk (Xn)n≥0, we characterize all possible ways of recycling these increments into a simple symmetric random walk (Yn)n≥0 adapted to the filtration of (Xn)n≥0. We study the long term behavior of a suitably normalized two-dimensional process ((Xn, Yn))n≥0. In particular, we provide necessary and sufficient conditions for the process to converge to a two-dimensional Brownian motion (possibly degenerate). We also discuss cases in which the limit is not Gaussian. Finally, we provide a simple necessary and sufficient condition for the ergodicity of the recycling transformation, thus generalizing results from Dubins and Smorodinsky (1992) and Fujita (2008), and solving the discrete version of the open problem of the ergodicity of the general Lévy transformation (see Mansuy and Yor, 2006).

Original languageEnglish
Article number92
Number of pages22
JournalElectronic Journal of Probability
Publication statusPublished - 2022


  • ergodicity
  • functional limit theorems
  • long memory
  • Lévy transformation
  • random walks

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