Backbone decomposition for continuous-state branching processes with immigration

In the spirit of Duquesne and Winkel (2007) and Berestycki et al. (2011), we show that supercritical continuous-state branching process with a general branching mechanism and general immigration mechanism is equivalent in law to a continuous-time Galton-Watson process with immigration (with Poissonian dressing). The result also helps to characterise the limiting backbone decomposition which is predictable from the work on consistent growth of Galton-Watson trees with immigration in Cao and Winkel (2010).     

Stochastic equations for two-type continuous-state branching processes with immigration

A two-dimensional stochastic integral equation system with jumps is studied. We first prove its unique weak solution is a two-type continuous-state branching process with immigration. Then the comparison property of the solution is established. These results imply the existence and uniqueness of the strong solution of the stochastic equation system.     

Parameter estimation in two-type continuous-state branching processes with immigration

We study the parameter estimation of two-type continuous-state branching processes with immigration based on low frequency observations at equidistant time points. The ergodicity of the processes is proved. The estimators are based on the minimization of a sum of squared deviation about conditional expectations. We also establish the strong consistency and central limit theorems of the conditional least squares estimators and the weighted conditional least squares estimators of the drift and diffusion coefficients based on low frequency observations.     

Stochastic equations for two-type continuous-state branching processes with immigration and competition

A class of two-type continuous-state branching processes with immigration and competition is constructed as the solution of a jump-type stochastic integral equation system. We first show that the stochastic equation system has a pathwise unique non-negative strong solution and then prove the comparison property of the solution.     

Extinction properties of multi-type continuous-state branching processes

Recently in Barczy et al. (2015), the notion of a multi-type continuous-state branching process (with immigration) having $d$-types was introduced as a solution to an $d$-dimensional vector-valued SDE. Preceding that, work on affine processes, originally motivated by mathematical finance, in Duffie et al. (2003) also showed the existence of such processes. See also more recent contributions in this direction due to Gabrielli and Teichmann (2014) and Caballero and Pérez Garmendia (2017). Older work on multi-type continuous-state branching processes is more sparse but includes Watanabe (1969) and Ma (2013), where only two types are considered. In this paper we take a completely different approach and consider multi-type continuous-state branching process, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms. In the spirit of Engländer and Kypriano (2004) we explore their extinction properties and pose a number of open problems.     

Markov branching processes with instantaneous immigration

Summary Markov branching processes with instantaneous immigration possess the property that immigration occurs immediately the number of particles reaches zero, i.e. the conditional expectation of sojourn time at zero is zero. In this paper we consider the existence and uniqueness of such a structure. We prove that if the sum of the immigration rates is finite then no such structure can exist, and we provide a necessary and sufficient condition for existence for the case in which this sum is infinite. Study of the uniqueness problem shows that for honest processes the solution is unique.     

Supercritical age-dependent branching processes with immigration

This paper considers a population process where individuals reproduce according to an age-dependent branching process and immigrants enter the population at the event epochs of an ergodic point process. A limit theorem is proven for what corresponds to the supercritical case, and the limit random variable is investigated.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

Sevastyanov branching processes with non-homogeneous Poisson immigration

Sevastyanov age-dependent branching processes allowing an immigration component are considered in the case when the moments of immigration form a non-homogeneous Poisson process with intensity r(t). The asymptotic behavior of the expectation and of the probability of non-extinction is investigated in the critical case depending on the asymptotic rate of r(t). Corresponding limit theorems are also proved using different types of normalization. Among them we obtained limiting distributions similar to the classical ones of Yaglom (1947) and Sevastyanov (1957) and also discovered new phenomena due to the non-homogeneity.     

The sector constants of continuous state branching processes with immigration

Continuous state branching processes with immigration are studied. We are particularly concerned with the associated (non-symmetric) Dirichlet form. After observing that gamma distributions are only reversible distributions for this class of models, we prove that every generalized gamma convolution is a stationary distribution of the process with suitably chosen branching mechanism and with continuous immigration. For such non-reversible processes, the strong sector condition is discussed in terms of a characteristic called the Thorin measure. In addition, some connections with notion from non-commutative probability theory will be pointed out through calculations involving the Stieltjes transform.     

On sequential estimation for branching processes with immigration

Consider a Galton–Watson process with immigration. The limiting distributions of the nonsequential estimators of the offspring mean have been proved to be drastically different for the critical case and subcritical and supercritical cases. A sequential estimator, proposed by Sriram et al. (Ann. Statist. 19 (1991) 2232), was shown to be asymptotically normal for both the subcritical and critical cases. Based on a certain stopping rule, we construct a class of two-stage estimators for the offspring mean. These estimators are shown to be asymptotically normal for all the three cases. This gives, without assuming any prior knowledge, a unified estimation and inference procedure for the offspring mean.     

Small value probabilities for continuous state branching processes with immigration

We consider the small value probability of supercritical continuous state branching processes with immigration.From Pinsky(1972) it is known that under regularity condition on the branching mechanism and immigration mechanism,the normalized population size converges to a non-degenerate finite and positive limit W as t tends to infinity.We provide sharp estimate on asymptotic behavior of P(W≤ε) as ε→ 0+ by studying the Laplace transform of W.Without immigration,we also give a simpler proof for the small value probability in the non-subordinator case via the prolific backbone decomposition.