PreprintCC BY-NC-ND 4.0arXiv
p-Wasserstein distance and Cramér-type moderate deviations for the multi-dimensional elephant random walk
Abstract
In this paper, we establish p-Wasserstein bounds and Cramér-type moderate deviations for the multi-dimensional elephant random walk in the diffusive and critical regimes. Using a Gaussian convolution estimate with random covariance and the recursions for the conditional covariance, we obtain Wp,2(ℒ(T̂n),N(0,Id))≤ Ca,dpLn, where T̂n is normalized by its exact covariance and Lnasymp n−1/2log n,&a≤1/3, n−3(1/2−a),&1/3<a<1/2, (log n)−3/2,&a=1/2. Consequently, we obtain radial Cramér-type moderate deviations for x=o(Ln−1/3). We also determine the covariance bias under the uncorrected normalization and establish the corresponding Wasserstein bounds, lower bounds and moderate deviation estimates. The dimension and the model parameters are fixed.
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BibTeX
@misc{liang2026pwasserstein,
title = {{$p$-Wasserstein distance and Cramér-type moderate deviations for the multi-dimensional elephant random walk}},
author = {Shuyuan Liang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01443},
eprint = {2611.01443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01443},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - p-Wasserstein distance and Cramér-type moderate deviations for the multi-dimensional elephant random walk
AU - Shuyuan Liang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01443
UR - https://arxiv.org/abs/2611.01443
AB - In this paper, we establish p-Wasserstein bounds and Cramér-type moderate deviations for the multi-dimensional elephant random walk in the diffusive and critical regimes. Using a Gaussian convolution estimate with random covariance and the recursions for the conditional covariance, we obtain W_p,2(ℒ(T̂_n),N(0,I_d))≤ C_a,dpL_n, where T̂_n is normalized by its exact covariance and L_nasymp n^−1/2log n,&a≤1/3, n^−3(1/2−a),&1/3<a<1/2, (log n)^−3/2,&a=1/2. Consequently, we obtain radial Cramér-type moderate deviations for x=o(L_n^−1/3). We also determine the covariance bias under the uncorrected normalization and establish the corresponding Wasserstein bounds, lower bounds and moderate deviation estimates. The dimension and the model parameters are fixed.
ER -