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Comparison on the criticality parameters for two supercritical branching processes with immigration in random environments
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- Journal:
- Probability in the Engineering and Informational Sciences , First View
- Published online by Cambridge University Press:
- 08 October 2025, pp. 1-34
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- Article
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This paper considers two supercritical branching processes with immigration in different random environments, denoted by
$\{Z_{1,n}\}$ and 
$\{Z_{2,m}\}$, with criticality parameters µ1 and µ2, respectively. Under certain conditions, it is known that 
$\frac{1}{n} \log Z_{1,n} \to \mu_1$ and 
$\frac{1}{m} \log Z_{2,m} \to \mu_2$ converge in probability as 
$m, n \to \infty$. We present basic properties about a central limit theorem, a non-uniform Berry–Esseen’s bound, and Cramér’s moderate deviations for 
$\frac{1}{n} \log Z_{1,n} - \frac{1}{m} \log Z_{2,m}$ as 
$m, n \to \infty$. To this end, applications to construction of confidence intervals and simulations are also given.
