Large Deviation Rates 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, 2021, vol. 34, issue 1, 162-172

Abstract: Abstract Let $$\{X_n\}_0^{\infty }$$ { X n } 0 ∞ be a supercritical branching process with immigration with offspring distribution $$\{p_j\}_0^{\infty }$$ { p j } 0 ∞ and immigration distribution $$\{h_i\}_0^{\infty }.$$ { h i } 0 ∞ . Throughout this paper, we assume that $$p_0=0, p_j\ne 1$$ p 0 = 0 , p j ≠ 1 for any $$j\ge 1$$ j ≥ 1 , $$1 \varepsilon ), \ \ P\left( \left| \frac{X_{n+1}}{X_n}-m\right| >\varepsilon \Bigg |Y\ge \alpha \right) \end{aligned}$$ P ( Y n - Y > ε ) , P X n + 1 X n - m > ε | Y ≥ α for $$\varepsilon >0$$ ε > 0 and $$\alpha >0$$ α > 0 under various moment conditions on $$\{p_j\}_0^{\infty }$$ { p j } 0 ∞ and $$\{h_i\}_0^{\infty }.$$ { h i } 0 ∞ . It is shown that the rates are always supergeometric under a finite moment generating function hypothesis.

Keywords: Large deviation; Supercritical branching process; Immigration; Primary 60J27; Secondary 60J35 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-019-00968-z

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