@article{Wang2025, 
author = {Juan Wang and Chao Peng},
title = {Lower deviation probabilities for supercritical Markov branching processes with immigration},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {5},
pages = {10324-10339},
keywords = {lower deviation, supercritical, branching process, immigration, Cramér method},
url = {https://www.sciopen.com/article/10.3934/math.2025470},
doi = {10.3934/math.2025470},
abstract = {Let    {  Z  (  t  )  ;  t  ≥  0  } be a continuous-time supercritical branching process with immigration (MBPI) with the offspring mean    m  (  t  ). In this paper, we mainly research the lower deviation probabilities        P    (  Z  (  t  )  =      k    t    ) and        P    (  0  ≤  Z  (  t  )  ≤      k    t    ) with        k    t        /        e          m      (      t      )        →  0 as    t  →  ∞. Moreover, we present the local limit theorem and some related estimates of the MBPIs. For our proofs, we use the well-known Cramér method to prove the large deviation of the sum of independent variables to satisfy our needs.}
}