An improvement on the large deviations for longest runs in Markov chains

Large deviations for longest success runs L(n) in Markov chains have been previously studied in Liu and Yang (2018) and Liu and Zhu (2020) under a technical assumption p10 < p00 +p11, with pij denoting the transition probability from i to j. In this note, we prove that all the results in Liu and Yang (2018) and Liu and Zhu (2020) still hold even without such an assumption. The main step in the proof is to derive an improved global estimation for the distribution function of L(n) without this assumption, which might be of independent interest. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).