Critical Galton-Watson Process Research Articles

Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

Estimation of Probabilities of Large Deviations for a Critical Galton–Watson Process

Estimation of Probabilities of Large Deviations for a Critical Galton–Watson Process

Further results on the critical Galton-Watson process with immigration

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, …) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where bj, (j = 0, 1, …) is the probability that j individuals enter the population at each generation. Defining Xn (n = 0, 1, …) to be the population size at the n- th generation, it is known that {Xn} defines a Markov chain on the non-negative integers.

On the critical Galton-Watson process with immigration

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, l, …) of giving rise to j progeny in the following generation and in which there is an independent immigration component where bj (j = 0, l, …) is the probability that j individuals enter the population at each generation. Then letting Xn (n = 0, l, …) be the population size of the n-th generation, it is known (Heathcote [4], [51]) that {Xn} defines a Markov chain on the non-negative integers. Unless otherwise stated, we shall consider only those offspring and immigration distributions that make the Markov chain {Xn} irreducible and aperiodic.

An Explicit-Limit Theorem for the Critical Galton–Watson Process with Immigration

Summary For a critical Galton–Watson branching process with immigration, {Xn}, in which the offspring distribution has finite variance and the immigration distribution finite mean, it is shown that the random variables {Xn/n} approach a gamma distribution in law as n → ∞.

The Galton-Watson Process with Mean One and Finite Variance

The Galton-Watson Process with Mean One and Finite Variance