Asymptotic behaviour of critical controlled branching processes with random control functions

Abstract

In this paper, we investigate the asymptotic behaviour of controlled branching processes with random control functions. In a critical case, we establish sufficient conditions for both their almost-sure extinction and for their nonextinction with a positive probability. For some suitably chosen norming constants, we also determine different kinds of limiting behaviour for this class of processes.

Controlled branching processes with continuous time

“….}. Note that the case δ = 1 is an open problem, along with some other critical subclasses considered in Theorem 4 in [10].…”

Minimum disparity estimation in controlled branching processes

“…Therefore, the offspring mean and variance are m = σ 2 = 7, and τ m = θ 0 λ = 2.1 (see (A3) for definition). The parameter τ m is referred to as the asymptotic mean growth rate and is the threshold parameter of the model (see González, Molina, and del Puerto (2005b)). In practice, control functions, φ n (k), following Poisson distributions of parameters λk are appropriate to describe an environment with expected immigration or emigration depending on λ > 1 or < 1.…”

Recurrence and Transience of Near-Critical Multivariate Growth Models: Criteria and Examples

“…In the 1-dimensional case our model equation simplifies to the difference equation X n+1 = X n + g(X n ) + ξ n with some function g : R + → R. A number of examples can be put into this framework, among others e.g. population size dependent branching processes [8], [10], controlled branching processes [4], branching processes in random environment [2], or nonlinear stochastic trends [5]. Our main condition is…”