Critical Galton–Watson Processes with Overlapping Generations

A properly scaled critical Galton–Watson process converges to a continuous state critical branching process ξ ⁢ ( ⋅ ) \xi(\,{\cdot}\,) as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite-dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals ( ∫ 0 y ξ ⁢ ( y - u ) ⁢ d u γ \int_{0}^{y}\xi(y-u)\,du^{\gamma} , y ≥ 0 y\geq 0 ) with a pertinent γ ≥ 0 \gamma\geq 0 .

1035COVID-19: Boom or bust?

Background In the final weeks of 2019, a SARS-CoV-2 virus slipped furtively from animal to human in China. As of March 13, there have been 1,34,918 confirmed cases, out of which 4,990 is the death count. We are predicting extinction or explosion of the virus from the current realization of a Galton Watson process. Methods Based on the region wise reported number of cases, total was calculated. The observed offspring distribution was found by calculating the difference between the total number of cases in consecutive days. Hence the distribution modelled using Sequential Probability Ratio Tests (SPRT) to predict whether extinction or explosion will occur for the current realization of the process. Kolmogorov-Smirnov test was performed on the data to check the distribution of fit. Results We assume conservative approach of SPRT. The geometric distribution fits to the data taken from January 2020 to March 12, 2020. The SPRT on the offspring distribution predicts extinction of the disease if the number of cases reported on a new day are less than 58 then the disease will extinct, and will explode if more than 9,990 cases. Conclusions Our results show that if COVID-19 transmission is established, understanding the effectiveness of control measures in different settings will be crucial for understanding the likelihood that transmission can eventually be effectively mitigated. Key messages Our analysis highlights the value of recording individual cases and analyzing geographically heterogeneous data of COVID-19. Our results also have implications for estimation of transmission dynamics using the number of exported cases from a specific area.

ON ASYMPTOTIC RELATIONS FOR THE CRITICAL GALTON – WATSON PROCESS

Determining the asymptotics of the continuation probability for a Galton–Watson branching process is one of the most important problems in the theory of branching processes. This problem was solved by A.N. Kolmogorov (1938) in the case when the process starts with a single particle, and the classical result is obtained. A similar result for continuous branching processes was proved by B.A. Sevastyanov (1951). The next term in the expansion for continuous branching processes was obtained by V.M. Zolotarev (1957). The next term in the expansion for continuous branching processes in the critical case was obtained by V.P. Chistyakov (1957); the asymptotic expansion in the subcritical case under the condition of finiteness of the k-factorial moment was obtained by R. Mukhamedkhanova (1966). Asymptotic expansions for discrete branching processes in the subcritical and supercritical cases, provided that any m-factorial moment is finite, were obtained by S.V. Nagaev and R. Mukhamedkhanova (1966). In the critical case, the weak convergence of the conditional distribution of the quantity P(Z(n) > 0)Z(n) under the condition Z(n) > 0 to the exponential distribution was proved by A.M. Yaglom (1947) for processes starting with a single particle in the case of finiteness of the third moment of the number of generations. Subsequently, Spitzer, Kesten, and Ney (1966) proved this result under the condition that the second moment is finite. A similar result for branching processes with continuous parameters was established by V.M. Zolotarev (1957). In this paper, we study the asymptotics of the probability of continuation of the critical Galton-Watson process, starting with η particles. In addition, we prove an analogue of Yaglom’s theorem for critical Galton – Watson processes starting with a random number of particles.

A unifying approach to branching processes in a varying environment

Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

Skeletal stochastic differential equations for continuous-state branching processes

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

Consider a continuous time process {Yt=ZNt, t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1>δ are estimated for three types of positive δ.