PreprintCC BY 4.0HAL
Are random random walks normal?
Abstract
Given a symmetric simple random walk (Xn)n≥0, the family of all symmetric simple random walks (Yn)n≥0 adapted to the filtration of (Xn)n≥0 was studied in Collevecchio et al. (2022). In particular, the authors established necessary and sufficient conditions under which the suitably normalized two-dimensional process ((Xn, Yn))n≥0 converges weakly to a two-dimensional Brownian motion. When this occurs, we say that the random walk (Yn)n is normal (with respect to (Xn)n). In this paper, we investigate whether a “randomly selected” (Yn)n≥0 is normal. We consider a very general randomization procedure and look at both the quenched and annealed settings.
Related
BibTeX
@misc{hamza2026random,
title = {{Are random random walks normal?}},
author = {Kais Hamza and Laurent Tournier},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01658},
eprint = {2610.01658},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05786003v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Are random random walks normal?
AU - Kais Hamza
AU - Laurent Tournier
PY - 2026
DA - 2026/10/01
PB - HAL
DO - 10.48550/arxiv.2610.01658
UR - https://hal.science/hal-05786003v1
AB - Given a symmetric simple random walk (Xn)n≥0, the family of all symmetric simple random walks (Yn)n≥0 adapted to the filtration of (Xn)n≥0 was studied in Collevecchio et al. (2022). In particular, the authors established necessary and sufficient conditions under which the suitably normalized two-dimensional process ((Xn, Yn))n≥0 converges weakly to a two-dimensional Brownian motion. When this occurs, we say that the random walk (Yn)n is normal (with respect to (Xn)n). In this paper, we investigate whether a “randomly selected” (Yn)n≥0 is normal. We consider a very general randomization procedure and look at both the quenched and annealed settings.
ER -