Bootstrap random walks

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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}).

Original languageEnglish
Pages (from-to)1744-1760
Number of pages17
JournalStochastic Processes and their Applications
Volume126
Issue number6
DOIs
Publication statusPublished - 1 Jun 2016

Keywords

  • Functional limit theorem
  • Random walks

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