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Bootstrap random walks

Author

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  • Collevecchio, Andrea
  • Hamza, Kais
  • Shi, Meng

Abstract

Consider a one dimensional simple random walk X=(Xn)n≥0. We form a new simple symmetric random walk Y=(Yn)n≥0 by taking sums of products of the increments of X and study the two-dimensional walk (X,Y)=((Xn,Yn))n≥0. We show that it is recurrent and when suitably normalised converges to a two-dimensional Brownian motion with independent components; this independence occurs despite the functional dependence between the pre-limit processes. The process of recycling increments in this way is repeated and a multi-dimensional analog of this limit theorem together with a transience result are obtained. The construction and results are extended to include the case where the increments take values in a finite set (not necessarily {−1,+1}).

Suggested Citation

  • Collevecchio, Andrea & Hamza, Kais & Shi, Meng, 2016. "Bootstrap random walks," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1744-1760.
  • Handle: RePEc:eee:spapps:v:126:y:2016:i:6:p:1744-1760
    DOI: 10.1016/j.spa.2015.11.016
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    Cited by:

    1. Collevecchio, Andrea & Hamza, Kais & Liu, Yunxuan, 2019. "Invariance principle for biased bootstrap random walks," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 860-877.

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    Keywords

    Random walks; Functional limit theorem;

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