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

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.