Subcritical Galton-Watson Process Research Articles

Asymptotic behaviour of critical decomposable 2-type Galton–Watson processes with immigration

In this paper the asymptotic behaviour of a critical 2-type Galton–Watson process with immigration is described when its offspring mean matrix is reducible, in other words, when the process is decomposable. It is proved that, under second or fourth order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from a critical decomposable 2-type Galton–Watson process with immigration converges weakly. The limit process can be described using one or two independent squared Bessel processes and possibly the unique stationary distribution of an appropriate single-type subcritical Galton–Watson process with immigration. Our results complete and extend the results of Foster and Ney (1978) for some strongly critical decomposable 2-type Galton–Watson processes with immigration.

Moderate deviations for the total population arising from a nearly unstable sub-critical Galton-Watson process with immigration

In this paper, we consider the moderate deviation of the nearly unstable sub-critical Galton-Waston process with immigration for the centered total population arising. We discuss the main influence factors for the form of moderate deviation in different stochastic processes. Moreover, we compute the exact rate function in every different situation of MDP.

Large and Moderate Deviations for the Total Population Arising from a Sub-critical Galton–Watson Process with Immigration

In this paper, we provide the exact forms of large and moderate deviations for the empirical mean of population and the centered total population of a sub-critical branching process with immigration. The rate functions in our large and moderate deviations are explicitly identified. Our theorems also apply to the models of the integer-valued autoregression. In computing the generating function requested by Gartner-Ellis theorem, our treatment substantially relies on an algorithm specifically designed for the autoregressive structure of our models.

Improvement of weight computational statistical modeling via the transition to a subcritical Galton-Watson process

Improvement of weight computational statistical modeling via the transition to a subcritical Galton-Watson process

An unexpected connection between branching processes and optimal stopping

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Zn with offspring distribution Y, there exists a random variable X such that the probability P(Zn = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X1,…, Xn, where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

An unexpected connection between branching processes and optimal stopping

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Z n with offspring distribution Y, there exists a random variable X such that the probability P(Z n = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X 1,…, X n , where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

Precise estimates of presence probabilities in the branching random walk

In the subcritical speed area of a supercritical branching random walk, we prove that when the number of generations grows the probability of presence is asymptotically proportional to the corresponding expectation as in a subcritical Galton-Watson process. This improves a known result on the logarithm of this probability. The basic tools are a discrete version of the Feynman-Kac representation and large deviations.

Moments for stationary and quasi-stationary distributions of markov chains

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

Moments for stationary and quasi-stationary distributions of markov chains

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

A Limit Theorem on a Subcritical Galton-Watson Process with Immigration

For a sub-critical Galton-Watson process $\mathbf{X}$ with immigration, estimators have been studied by Pakes (1971) for the mean of the stationary distribution of $\mathbf{X}$, and by Nanthi (1979) for the offspring mean and the immigration mean of $\mathbf{X}$. These estimators have been shown by them to be asymptotically normal. In this paper it is shown that they have a joint limiting distribution which is singularly normal.

A time series approach to the study of the simple subcritical Galton–Watson process with immigration

The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson process X with immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems on X are proved leading to the construction of Quenouille-type goodness-of-fit tests for X. It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).

A time series approach to the study of the simple subcritical Galton–Watson process with immigration

The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson process X with immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems on X are proved leading to the construction of Quenouille-type goodness-of-fit tests for X. It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).

On a Schröder equation arising in branching processes

It is shown that a 1-1 correspondence exists between the possible Yaglom conditional limits when a subcritical Galton-Watson process is initiated with an arbitrary probability distribution and the invariant measures of the process. This is proven by an examination of the relevant Schroder and Abel functional equations.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

A condition for the extinction of a bounded branching process

A subcritical Galton-Watson process is investigated under the condition that the number of particles is bounded above (“superfluous” particles are annihilated). A necessary and sufficient condition is obtained for the probability of extinction of such a process to be equal to unity.