Shimi, I. (1970). A Generalized Age-Dependent Branching Process and Its Limit Distribution. The Egyptian Statistical Journal, 14(1), 13-32. doi: 10.21608/esju.1970.315803
I. N. Shimi. "A Generalized Age-Dependent Branching Process and Its Limit Distribution". The Egyptian Statistical Journal, 14, 1, 1970, 13-32. doi: 10.21608/esju.1970.315803
Shimi, I. (1970). 'A Generalized Age-Dependent Branching Process and Its Limit Distribution', The Egyptian Statistical Journal, 14(1), pp. 13-32. doi: 10.21608/esju.1970.315803
Shimi, I. A Generalized Age-Dependent Branching Process and Its Limit Distribution. The Egyptian Statistical Journal, 1970; 14(1): 13-32. doi: 10.21608/esju.1970.315803
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.
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