A Fluctuation Limit Theorem of Branching Processes with Immigration and Statistical Applications

where {ξ(k, j) : k, j = 1, 2, · · · } and {η(k) : k = 1, 2, · · · } are two independent families of i.i.d. random variables taking values in N := {0, 1, 2, · · · }. The distribution of ξ(k, j) is called the offspring distribution and the distribution of η(k) is called the immigration distribution. Let g(·) and h(·) be the generating functions of ξ(k, j) and η(k), respectively. It is easy to see that {y(k)} is a discrete-time Markov chain with values in N and one-step transition matrix P (i, j) given by