2015/08/12 by Andrea Collevecchio, Kaïs Hamza, Collevecchio, Andrea +4
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR
paper · pdf · doi:10.48550/arxiv.1508.02840
openalex publication_date 2015/08/12 · arxiv created 2015/08/17 · arxiv updated 2015/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\).