Related papers: A scaling limit theorem for controlled branching p…
We consider a multitype branching process with immigration in a random environment introduced by Key in [Ann. Probab. 15 (1987) 344--353]. It was shown by Key that, under the assumptions made in [Ann. Probab. 15 (1987) 344--353], the…
In this article we derive a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients under the condition that all partial sums of the series of…
In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem…
Population-size dependent branching processes (PSDBP) and controlled branching processes (CBP) are two classes of branching processes widely used to model biological populations that exhibit logistic growth. In this paper we develop…
Markov branching systems form a fundamental class of stochastic models that are extensively applied in biology, physics, finance, and other domains. These systems are distinguished by their continuous-time evolution and inherent branching…
We prove a general fluctuation limit theorem for Galton-Watson branching processes with immigration. The limit is a time-inhomogeneous OU type process driven by a spectrally positive Levy process. As applications of this result, we obtain…
We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. The models…
This paper deals into the long-term behavior of subordinated critical branching processes with migration. We focus on scenarios where emigration is the dominant factor and introduce additional randomness in timing through a subordination…
These notes were used in a short graduate course on branching processes the author gave in Beijing Normal University. The following main topics are covered: scaling limits of Galton--Watson processes, continuous-state branching processes,…
The standard small-time functional central limit theorem of semimartingales has been established in (Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications.…
We consider a random interval splitting process, in which the splitting rule depends on the empirical distribution of interval lengths. We show that this empirical distribution converges to a limit almost surely as the number of intervals…
The paper discusses the continuous-time Markov Branching Process allowing Immigration. We are considering a critical case for which the second moment of offspring law and the first moment of immigration law are possibly infinite. Assuming…
We present in this paper a new sufficient condition for the so-called Prokhorov-Skorokhod continuity of random processes. Our conditions will be formulated in the terms of metric entropy generated by three-dimensional distribution of the…
We give a concise self-contained presentation of known and new limit theorems for the one-type Markov branching processes with continuous time. The new streamlined proofs are based on what we call, the tail generating function approach. Our…
We extend existing connections between random walks, branching processes, and spatial branching processes, and their respective scaling limits, to include processes in dependent random environments. More specifically, we prove new scaling…
Consider a branching process $\{Z_n\}_{n\ge 0}$ with immigration in varying environment. For $a\in\{0,1,2,...\},$ let $C=\{n\ge0:Z_n=a\}$ be the collection of times at which the population size of the process attains level $a.$ We give a…
In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatial-dependent branching mechanisms. These are generalizations of the spatial central limit theorems…
We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring…
According to the Dudley-Wichura extension of the Skorohod representation theorem, convergence in distribution to a limit in a separable set is equivalent to the existence of a coupling with elements converging a.s. in the metric. A density…
The Chinese restaurant process is a basic sequential construction of consistent random partitions. We consider random point measures describing the composition of small blocks in such partitions and show that their scaling limit is given by…