Lower deviations for branching processes with immigration

TL;DR

This paper investigates the asymptotic behavior of lower deviation probabilities in critical branching processes with immigration, focusing on the probabilities that the process stays below or equals a certain threshold as the process size grows.

Contribution

It provides new asymptotic results for lower deviation probabilities in critical branching processes with immigration, emphasizing the impact of moment conditions.

Findings

Asymptotic formulas for P(Y_n ≤ k) and P(Y_n = k) as n, k → ∞ with k = o(n)

Clarification of the role of moment conditions in local limit theorems for Y_n

Extension of previous results by Mellein on local limit theorems

Abstract

Let $\{Y_{n}$, $n \geq 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper, we study lower deviation probabilities for $Y_{n}$. More precisely, assuming that $k,n \to \infty$ such that $k=o\left(n \right)$, we investigate the asymptotics of $\mathbf{P}\left(Y_{n} \leq k \right)$ and $\mathbf{P}\left(Y_{n} = k \right)$. Our results clarify the role of the moment conditions in the local limit theorem for $Y_n$ proven by Mellein.