超临界多类型带 immigration 分支过程的大偏差率
概率论
2025-08-22 v1
摘要
设 { X n } n ≥ 0 \{X_n\}_{n\geq0} { X n } n ≥ 0 p p p p ≥ 2 p\geq2 p ≥ 2 M M M M M M ρ \rho ρ M M M v \boldsymbol{v} v u \boldsymbol{u} u ρ > 1 \rho>1 ρ > 1 Y n = ρ − n [ u ⋅ X n − ρ n + 1 − 1 ρ − 1 ( u ⋅ λ ) ] Y_n=\rho^{-n}\Big[\boldsymbol{u}\cdot X_n -\frac{\rho^{n+1}-1}{\rho-1}( \boldsymbol{u}\cdot \boldsymbol{\lambda})\Big] Y n = ρ − n [ u ⋅ X n − ρ − 1 ρ n + 1 − 1 ( u ⋅ λ ) ] λ \boldsymbol{\lambda} λ Y n Y_n Y n n → ∞ n\rightarrow\infty n → ∞ Y Y Y n → ∞ n\rightarrow\infty n → ∞ Y n Y_n Y n 0 0 0 P i ( ∣ l ⋅ X n + 1 1 ⋅ X n − l ⋅ ( X n M ) 1 ⋅ X n ∣ > ε ) , P i ( ∣ l ⋅ X n 1 ⋅ X n − l ⋅ v 1 ⋅ v ∣ > ε ) , P ( ∣ Y n − Y ∣ > ε ) P_i\Big(\Big|\frac{\boldsymbol{l}\cdot X_{n+1}}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot(X_nM)}{\textbf{1}\cdot X_n}\Big|>\varepsilon\Big),P_i\Big(\Big|\frac{\boldsymbol{l}\cdot X_n}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot\boldsymbol{v}}{\textbf{1}\cdot \boldsymbol{v} }\Big|>\varepsilon\Big),P\Big(\Big|Y_n-Y\Big|>\varepsilon\Big) P i ( 1 ⋅ X n l ⋅ X n + 1 − 1 ⋅ X n l ⋅ ( X n M ) > ε ) , P i ( 1 ⋅ X n l ⋅ X n − 1 ⋅ v l ⋅ v > ε ) , P ( Y n − Y > ε ) ε > 0 , i = 1 , ⋯ , p \varepsilon>0, i=1,\cdots,p ε > 0 , i = 1 , ⋯ , p 1 = ( 1 , ⋯ , 1 ) \textbf{1}=(1,\cdots,1) 1 = ( 1 , ⋯ , 1 ) l ∈ R p \boldsymbol{l}\in\mathbb{R}^p l ∈ R p Y ≥ α Y\geq\alpha Y ≥ α α > 0 \alpha>0 α > 0
引用
@article{arxiv.2508.15428,
title = {Large deviation rates for supercritical multitype branching processes with immigration},
author = {Liuyan Li and Junping Li},
journal= {arXiv preprint arXiv:2508.15428},
year = {2025}
}