Lower deviations for general supercritical branching process with immigration

For a supercritical branching process with immigration (Z_n), a sequence of constant c_n could be used to describe the growth rate of the process．The asymptotic behavior of P(Z_n=k_n) (k_n=o(c_n)) is called the lower deviation probability of Z_n ．In this paper, under EZ_1 \ln Z_1=\infty , first, a local limit theorem of Z_n is proved．Then in the Schröder and Böttcher cases, the lower deviation probability P(Z_n=k_n) is discussed, which improves and generalizes the corresponding results in the literature．