Consider a (2, 1) random walk in an i.i.d. random environment, whose environment involves certain parameter. We construct an M-estimator for the environment parameter which can be written as functionals of a multitype branching process with immigration in a random environment (BPIRE). Because the offspring distributions of the involved multitype BPIRE are of the linear fractional type, the limit invariant distribution of the multitype BPIRE can be computed explicitly. As a result, we get the weak consistency of the M-estimator. Our result is a generalization of Comets et al. [Stochastic Process. Appl. 2014, 124, 268-288].

Metastasis, the spread of cancer cells from a primary tumor to secondary location(s) in the human organism, is the ultimate cause of death for the majority of cancer patients. Although studied for more than 180 years, increasing efforts in recent years have significantly contributed to a better understanding of this aspect of tumor development. Adding to this understanding, our current paper proposes a multi-type branching process model, through which various topics, such as extinction of cancer cell clones, occurrence of successful mutants, and immediate risk of escaping extinction, are investigated. More specifically, the relationship between the lifespan distribution of cancer cells with subcritical reproduction and the waiting time until the occurrence of a mutant (having supercritical reproduction) cancer cell, escaping extinction, is highlighted. The theoretical studies formulated in this paper may allow our model to be tailored to real data available for particular kinds of cancer and chemotherapy.

Branching processes in random environments arise in a variety of applications such as biology, finance, and other contemporary scientific areas. Motivated by these applications, this article investigates the problem of ancestral inference. Specifically, the article develops point and interval estimates for the mean number of ancestors initiating a branching process in i.i.d. random environments and establishes their asymptotic properties when the number of replications diverges to infinity. These results are then used to quantitate the number of DNA molecules in a genetic material using data from polymerase chain reaction experiments. Numerical experiments and data analyses are included to support the proposed methods. An R software package for implementing the methods of this manuscript is also included.

Branching processes are widely used to model the viral epidemic evolution. For more adequate investigation of viral epidemic modeling, we suggest to apply branching processes with transport of particles usually called branching random walks (BRWs). This allows to investigate not only the number of particles (infected individuals), but also their spatial spread. We consider two models of continuous-time BRWs on a multidimensional lattice in which the transport of infected individuals is described by a symmetric random walk on a multidimensional lattice whereas the processes of birth and death of infected individuals are represented by a continuous-time Bienayme-Galton-Watson processes at the lattice points (branching sources). A special attention is paid to the properties of branching random walks with one branching source on the lattice and finitely or infinitely many initial particles. We show that there exists a kind of duality between the branching random walk with a finite number of initial particles and the branching random walk with an infinite number of initial particles, which is associated with the possibility of their twofold description. The fact of duality is useful from the biological point of view. Each of the models can be considered taking into account the vaccination process. We suppose the vaccination to be a proportion of immune individuals in the population, who are resistant to disease. For simplicity, in all our BRW models, we assume that the vaccination process does not depend on time, what allows to investigate spatial properties of viral evolution.

In this paper, we consider a variant of a discrete time Galton-Watson Branching Process in which an individual is allowed to survive for more than one (but finite) number of generations and may also give birth to offsprings more than once. We model the process using multitype branching process and derive conditions on the mean matrix that determines the long-run behavior of the process. Next, we analyze the distribution of the number of forefathers in a given generation. Here, number of forefathers of an individual is defined as all the individuals since zeroth generation who have contributed to the birth of the individual under consideration. We derive an exact expression for expected number of individuals in a given generation having a specified number of forefathers. Using this exact expression, we provide a detailed analysis for a simple illustrative case. Some interesting insights and possible applications are also discussed.

The weighted branching process is a natural extension of branching processes. Branching processes count only individuals, weighted branching processes give every individual an abstract weight. Mathematically, we face a (random) dynamical system indexed by a tree. We give an overview of research into this directions, including and unifying many examples of this structure, like Biggins branching random walk and Mandelbrot cascades. WBP are closely connected to stochastic fixed-point equations, fractals and have many applications in genetics, computer science and algorithms. The most advanced example we present is the Quicksort process.

In Jacob et al. [A General Class of Population-Dependent Two-Sex Processes with Random Mating. Bernoulli 2017, 23, 1737-1758], a new class of two-sex branching processes in discrete time was introduced. These processes present the novelty that, in each generation, mating between females and males is randomly governed by a set of Bernoulli distributions allowing polygamous behavior with only perfect fidelity on the part of female individuals. Moreover, mating as well as reproduction can be influenced by the number of females and males in the population. In that article, the authors study conditions leading to the almost sure extinction or to the possible survival with positive probability of such processes. In this work, we continue the research about this class of two-sex processes by investigating the rate of growth of the population in case of survival. In particular, we provide conditions for a simultaneous geometric growth of the number of females and males and conditions for the geometric growth of females with a stable population of males.

We prove the existence and pathwise uniqueness of the solution to a stochastic integral equation driven by Poisson random measures based on Kuznetsov measures for a continuous-state branching process. That gives a direct construction of the sample path of a continuous-state branching process with dependent immigration. The immigration rates depend on the population size via some functions satisfying a Yamada-Watanabe type condition. We only assume the existence of the first moment of the process. The existence of excursion law for the continuous-state branching process is not required. By special choices of the ingredients, we can make changes in the branching mechanism or construct models with competition.

This note provides a factorization of a Lévy pocess over a phase-type horizon τ given the phase at the supremum, thereby extending the Wiener-Hopf factorization for τ exponential. One of the factors is defined using time reversal of the phase process. It is shown that there are a variety of time-reversed representations, all yielding the same factor. Consequences of this are discussed and examples provided. Additionally, some explicit formulas for the joint law of the supremum and the terminal value of the process at τ are given.

We consider critical Sevastyanov branching processes with immigration at random time-points generated by a Poisson random measure with a local intensity for some slowly varying function The asymptotic behavior of the probability of non-extinction is studied and conditional limiting distributions of the processes with proper normalization are obtained.

We investigate conditions for survival in the L 1 -norm sense of the Log-Laplace equations of a class of Markov branching processes with values in the space of Radon measures on a locally compact space D. We apply our results to certain -valued superprocesses with -branching.

We give sufficient conditions on the offspring, the initial and the immigration distributions under which a second-order Galton-Watson process with immigration is regularly varying.