Lower deviation probabilities for supercritical multi-type Galton--Watson processes

2025/06/30 by Jiangrui, Tan
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.23647

Abstract

This paper provides a detailed analysis of the lower deviation probability properties for a d-type (d>1) Galton--Watson (GW) process \Z_n=(Z_n⁽ⁱ⁾)_{1≤ i≤ d};n≥0\ in both Schröder and Böttcher cases. We establish explicit decay rates for the following probabilities: ℙ(Z_n=k_n),~ ℙ(|Z_n|≤ k_n), ~ℙ(Z⁽ⁱ⁾_n=k_n)~~and~~ℙ(Z⁽ⁱ⁾_n≤ k_n), 1≤ i ≤ d, respectively, where k_n∈ℤ₊^d, |k_n|=o(c_n), k_n=o(c_n) and c_n characterizes the growth rate of Z_n. These results extend the single-type lower deviation theorems of Fleischmann and Wachtel (Ann. Inst. Henri Poincaré Probab. Statist.43 (2007) 233-255), thereby paving the way for analysis of precise decay rates of large deviations in multi-type GW processes.

Citations

• Lower deviation and moderate deviation probabilities for maximum of a branching random walk
• On large deviation rates for sums associated with Galton-Watson processes
• Lower large deviations for supercritical branching processes in random environment
• Lower deviation probabilities for supercritical Galton-Watson processes

Related

• Lower deviation probabilities for supercritical Galton-Watson processes
• On large deviation rates for sums associated with Galton-Watson processes
• Large deviations for sums defined on a Galton-Watson process
• Lower large deviations for supercritical branching processes in random environment
• Local limit theory and large deviations for supercritical Branching processes