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Moderate deviations for the Maki--Thompson rumour model
Abstract
We consider the classical Maki--Thompson rumour model with n initially ignorant individuals and one initial spreader. The final number of ignorants, divided by n, is known to satisfy the law of large numbers, the central limit theorem and the large deviation principle. We establish a moderate deviation principle for fluctuations of this proportion of order bn/√n, where bn→∞ and bn2=o(log n). The proof rests on the exact final-size distribution, sharp asymptotics for the associated automata numbers, and a uniform point probability expansion at the moderate deviation scale. Numerical comparisons of both tails illustrate the intermediate scale and the effect of early extinction on upper tail probabilities.
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BibTeX
@misc{wang2026moderate,
title = {{Moderate deviations for the Maki--Thompson rumour model}},
author = {Shaochen Wang and Guangyu Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.08629},
eprint = {2605.08629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08629},
}
RIS
TY - UNPB
TI - Moderate deviations for the Maki--Thompson rumour model
AU - Shaochen Wang
AU - Guangyu Yang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2605.08629
UR - https://arxiv.org/abs/2605.08629
AB - We consider the classical Maki--Thompson rumour model with n initially ignorant individuals and one initial spreader. The final number of ignorants, divided by n, is known to satisfy the law of large numbers, the central limit theorem and the large deviation principle. We establish a moderate deviation principle for fluctuations of this proportion of order b_n/√n, where b_n→∞ and b_n^2=o(log n). The proof rests on the exact final-size distribution, sharp asymptotics for the associated automata numbers, and a uniform point probability expansion at the moderate deviation scale. Numerical comparisons of both tails illustrate the intermediate scale and the effect of early extinction on upper tail probabilities.
ER -