Harmonic Moments and Large Deviations for the Markov Branching Process with Immigration

Liuyan Li () and Junping Li ()
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Liuyan Li: Guangdong University of Science & Technology
Junping Li: Guangdong University of Science & Technology

Journal of Theoretical Probability, 2024, vol. 37, issue 2, 1397-1416

Abstract: Abstract Let $$\{X_t;t\ge 0\}$$ { X t ; t ≥ 0 } be a Markov branching process with immigration, in which each particle has exponential lifetime distribution with parameter a and offspring law $$\{p_k;k=0,1,2,\ldots \}$$ { p k ; k = 0 , 1 , 2 , … } . The time interval of immigration follows an exponential distribution with parameter $$\theta $$ θ . Let $$p_0=0$$ p 0 = 0 and $$\lambda :=a(\sum _{k=1}^{\infty }kp_k-1) 0\right\} $$ X t + s / X t ; t ≥ 0 , s > 0 by studying the asymptotic behavior of harmonic moments $$E[X_t^{-r}]$$ E [ X t - r ] for $$r>0$$ r > 0 . We obtain that there is a “phase transition" in rates depending on whether $$\lambda r-a-\theta $$ λ r - a - θ is less than, equal to or greater than 0.

Keywords: Harmonic moments; Large deviations; Markov branching process; Immigration; 60F10; 60J80 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01280-7

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