The Local Limit Theorem for Supercritical Branching Processes with Immigration

Liuyan Li () and Junping Li ()
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Liuyan Li: Central South University
Junping Li: Central South University

Journal of Theoretical Probability, 2023, vol. 36, issue 1, 331-347

Abstract: Abstract Let $$\{X_n\}_{n\ge 0}$$ { X n } n ≥ 0 be a supercritical Galton–Watson branching process with immigration initiated by a random number $$X_0$$ X 0 of initial individuals, with offspring distribution $$\{p_j\}_{j\ge 0}$$ { p j } j ≥ 0 and immigration distribution $$\{h_j\}_{j\ge 0}$$ { h j } j ≥ 0 . Throughout the paper, we assume that $$p_0=0$$ p 0 = 0 and $$h_0>0$$ h 0 > 0 . Let $$V_n=X_n/c_n$$ V n = X n / c n and $$\lim \limits _{n\rightarrow \infty }V_n=V$$ lim n → ∞ V n = V , where $$c_n$$ c n is the well-known sequence of constants describing the growth of $$X_n$$ X n . In this paper, we obtain a local limit theorem for the process $$\{X_n\}_{n\ge 0}$$ { X n } n ≥ 0 in both the case $$p_1>0$$ p 1 > 0 and the case $$p_1=0$$ p 1 = 0 .

Keywords: Local limit theorems; Supercritical; Branching processes; Immigration; Primary 60J27; Secondary 60J35 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01171-3

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