Projects per year
Abstract
Given the increments of a simple symmetric random walk (X_{n})_{n≥0}, we characterize all possible ways of recycling these increments into a simple symmetric random walk (Y_{n})_{n≥0} adapted to the filtration of (X_{n})_{n≥0}. We study the long term behavior of a suitably normalized twodimensional process ((X_{n}, Y_{n}))_{n≥0}. In particular, we provide necessary and sufficient conditions for the process to converge to a twodimensional 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 language  English 

Article number  92 
Number of pages  22 
Journal  Electronic Journal of Probability 
Volume  27 
DOIs  
Publication status  Published  2022 
Keywords
 ergodicity
 functional limit theorems
 long memory
 Lévy transformation
 random walks
Projects
 1 Finished

Random walks with long memory
Collevecchio, A., Garoni, T., Hamza, K., Cotar, C. & Tarres, P.
Australian Research Council (ARC)
1/05/18 → 1/05/22
Project: Research