The Branching Process With Random Environments Adapted To A Family Of σ-Fields

Galton and Watson set up a kind of new random process model, i.e. classical branching process, in discussing the question of British noble's surname succession and the question of withering away of the pedigree. However, the homogeneity assumption that all individuals independently reproduce their children following the same probability distribution law looks contradiction with procreation course in nature where the interaction between individuals and influence by other factors must exist, which makes the applications of the classical branching process get certain restriction. This defect was remedied by the branching processes in random environments such as with independent identically distributed random environments proposed by Smith and Wilkinson and the branching process with a stationary ergodic process as random environments suggested by Athreya and Karlin.Only a special kind of branching process model presented by Jagers and Lu (2002) is consider in this paper, where the random environments process {ζ_n} is adapted to a family of or-fields, say {B_n}. Here B_n can include the historical of the branching process itself and other influence of environments. Particularly,B_n can be defined as the inducedσ-field inΩproduced by random variables of Z₀,…,z_n,ζ₁,…ζ_n, i.e. B_n=σ(Z₀,…,Z_n,ζ₁,…,ζ_n). Furthermore, {ζ₁} can be supposed to be independent identically distributed and their values are notdependent on {Z_k} Then, there must exist a B_R²ⁿ⁺¹-measurable function g_n such thatζ_n=g_n(Z₀,…,Z_n,ζ₁,…,ζ_n) for every n=0,1,2,…The main works made in this kind of situation of the paper are:1. Probability generating functions of Z_n were discussed and the fact that no longer like the Smith-Wilkinson branching process in random environments and the Athreya-Karlin branching process in random environments in general the probability generating functions of Z_n can not be expressed by some expectation for an iteration ofφ_ζ0,…,φ_ζn-1 showed with the counterexamples.2. Provided some martingale expresses of the process {z_n}.3. Provided the sufficient condition of the extinction probability of process {z_n}equal to 1.4. Provided the sufficient condition of the extinction probability of process {Z_n} smaller than 1.5. Discussed the extinction probability of branching process in deteriorated random environments.