A Generalized Age-Dependent Branching Process and Its Limit Distribution

An age-dependent continuous parameter branching stochastic process Xₙ(t) is considered, where Xₙ (t) is the number of particles in the population at time t and N is the initial size of the population. If the probabilities of splits depend on N. in a way that will be made precise in the text, then, for fixed t. as N tends to infinity a limiting distribution of the stochastic processes Xₙ (t)__N is obtained, and shown to be the distribution of a continuous parameter stochastic process with independent increments X (t), whose distribution will be determined. To establish this we need to prove that for t₁< t₂ <... <tₙ the distribution of Xₙ (tᵢ) -- X(tᵢ₋₁) converges to the distribution of X(tᵢ) --X(tᵢ₋₁), and  Xₙ (t₂) --  Xₙ (t₁).  Xₙ(t₃) --  Xₙ(t₂),….,  Xₙ (tₙ) -- Xₙ ( tₙ₋₁) are independent in the limit. The limiting distribution of Xₙ (t) - N gives an approximation to the distribution of Xₙ (t) for any fixed t and large N.