where the empty sum is considered to be 0, N is a positive integer, and {Xnj :n = 0, 1, . . . ; j = 1, 2, . . . } and {φn(k):n, k = 0, 1, . . . } are independent sets of nonnegative-integer-valued random variables defined on the same probability space. The variables Xnj are independent and identically distributed with common probability law {pk}k≥0, pk := P(X01 = k), k = 0, 1, . . . , called the offspring probability distribution, and for n = 0, 1, . . . , {φn(k)}k≥0 are independent stochastic processes such that φn(k), n = 0, 1, . . . , are identically distributed. Let us denote by m := E[X01], σ := Var[X01], e(k) := E[φ0(k)], and ν(k) := Var[φ0(k)], k = 0, 1, . . . , the mean and variance (assumed finite) of the offspring probability distribution and the control random variables, respectively. From (1), it is easily verified that {Zn}n≥0 is a homogeneous Markov chain with state space on the nonnegative integers. For simplicity, we shall consider that the positive integers form a class of communicating and aperiodic states, and we will assume that P(φ0(0) = 0) = 1 and at least one of the following conditions holds: (i) p0 > 0; (ii) P(φ0(k) = 0) > 0, k = 1, 2, . . . . Under these assumptions, it was proved in [14] that the positive states are transient and, as a consequence, it is verified that P(Zn → 0) +P(Zn → ∞) = 1. (2)