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 branching process is subcritical in the sense that it converges to a proper limit law. We complement this result by a strong law of large numbers and a central limit theorem for the partial sums of the process. In addition, we study the asymptotic behavior of oscillations of the branching process, that is, of the random segments between successive times when the extinction occurs and the process starts again with the next wave of the immigration.
In this short note we obtain necessary and sufficient conditions for the existence of the moments of the stationary distribution of a subcritical multitype Galton-Watson process with immigration.
For multitype branching processes with immigration evolving in a random environment and producing a final product we find the tail distribution of the size of the final product accumulated in the system for a life period. Using this result we investigate the tail distribution of the busy periods of the branching type polling systems with random service disciplines and random positive switch-over times
Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton-Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton-Watson processes with immigration as well.
In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a generalized Polya process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the renormalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient moment analysis when there are only two types of particles is provided. AMS 2000 subject classi…
We provide a simple forest model to encode the genealogical structure of a multitype Galton-Watson process with immigration. We provide two encodings of these forests by stochastic processes. We show, under appropriate conditions, the depth-first encodings of each particular type converge to a solution to a system of stochastic integral equations involving height processes perturbed by functionals of their local times. The forest picture allows us to extend the Ray-Knight theorem and show that local time of the solution to the system of equations form a multitype continuous state branching pro…
We first study the convergence of solutions of a system of F-KPP equations related to irreducible multitype branching Brownian motions with Heaviside-type initial conditions to traveling wave solutions. Then we apply this convergence result to prove that the extremal processes of irreducible multitype branching Brownian motions converges weakly to a cluster point process.
We introduce a Poissonization method to study the coalescent structure of uniform samples from branching processes. This method relies on the simple observation that a uniform sample of size $k$ taken from a random set with positive Lebesgue measure may be represented as a mixture of Poisson samples with rate $λ$ and mixing measure $k \mathrm{d} λ/ λ$. We develop a multitype analogue of this mixture representation, and use it to characterise the coalescent structure of multitype continuous-state branching processes in terms of random multitype forests. Thereafter we study the small time asympt…
By the methods of multitype branching processes in random environment counted by random characteristics we study the tail distribution of busy periods and some other characteristics of the branching type polling systems in which the service disciplines, input parameters and service time distributions are changing in a random manner.
We study the asymptotic behavior of the probability of non extinction of a weakly subcritical multitype branching process in iid random environments. Under suitable assumptions, the survival probability is of order of $ρ^n n ^{-3/2}$ for some $ρ\in (0, 1)$ to specify.
We consider multitype branching processes arising in the study of random laminations of the disk. We classify these processes according to their subcritical or supercritical behavior and provide Kolmogorov-type estimates in the critical case corresponding to the random recursive lamination process of [1]. The proofs use the infinite dimensional Perron-Frobenius theory and quasi-stationary distributions.
We study the asymptotic behaviour of the survival probability of a multitype branching process in random environment. The class of processes we consider here corresponds, in the one-dimensional situation, to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.
Motivated by tumor growth models we establish conditions for the $R-$positivity of Markov processes and positive matrices. We then apply them to obtain the asymptotic behavior of the tumors sizes in the supercritical regime.
In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with non-local branching mechanisms under a martingale change of measure. As an application of this decomposition, we obtain a necessary and sufficient condition (called the $L\log L$ criterion) for the limit of the fundamental martingale to be non-degenerate. This result complements the related results for superprocesses with purely local branching mechanisms and in Kyprianou and Palau (2016) for super Markov chains.
By extending the breadth first search algorithm to any d-type critical or subcritical irreducible branching forest, we show that such forests may be encoded through d independent, integer valued, d-dimensional random walks. An application of this coding together with a multivariate extension of the Ballot Theorem which is proved here, allow us to give an explicit form of the law of the total progeny, jointly with the number of subtrees of each type, in terms of the offspring distribution of the branching process. We then apply these results to some enumeration formulas of multitype forests wit…
The forest of mutations associated to a multitype branching forest is obtained by merging together all vertices of its clusters and by preserving connections between them. We first show that the forest of mutations of any mulitype branching forest is itself a branching forest. Then we give its progeny distribution and describe some of its crucial properties in terms the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to infinity and when the number of roots tends to infinity. The continuous time case…
We provide a simple set of sufficient conditions for the weak convergence of discrete Galton-Watson branching processes with immigration to continuous time and continuous state branching processes with immigration.
Motivated by the stochastic Lotka-Volterra model, we introduce discrete-state interacting multitype branching processes. We show that they can be obtained as the sum of a multidimensional random walk with a Lamperti-type change proportional to the population size; and a multidimensional Poisson process with a time-change proportional to the pairwise interactions. We define the analogous continuous-state process as the unique strong solution of a multidimensional SDE. We prove that the scaling limits of the discrete-state process correspond to its continuous counterpart. In addition, we show th…
In this paper we consider a triangular array of branching processes with non-stationary immigration. We prove a weak convergence of properly normalized branching processes with immigration to deterministic function under assumption that immigration is rowwise $ψ-$mixing and the offspring mean tends to its critical value 1, immigration mean and variance controlled by regularly varying functions. Moreover, we obtain a fluctuation limit theorem for branching process with immigration when immigration is $m-$dependent where $m$ may tend to infinity with the row index at a certain rate. In this case…
The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson ran- dom measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.
A multitype continuous-state branching process (MCSBP) ${\rm Z}=({\rm Z}_{t})_{t\geq 0}$, is a Markov process with values in $[0,\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\mathbb{R}^d$-valued spectrally positive Lévy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing…
In the spirit of Duqesne 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 equal in law to a continuous-time Galton Watson process with immigration with Poissonian dressing. The result also characterises the limiting backbone decomposition which is predictable from the work on consistent growth of Galton-Watson trees with immigration in Cao and Winkel (2010).
Branching processes and Fleming-Viot processes are two main models in stochastic population theory. Incorporating an immigration in both models, we generalize the results of Shiga (1990) and Birkner et al. (2005) which respectively connect the Feller diffusion with the classical Fleming-Viot process and the alpha-stable continuous state branching process with the Beta(2-alpha, alpha)-generalized Fleming-Viot process. In a recent work, a new class of probability-measure valued processes, called M-generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called …
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 some asymptotic estimates for the conditional least-squares estimator of the offspring means and variances of the offspring and immigration distributions.
Using the annealed approach we investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in i.i.d. random environment. We show under rather general assumptions on the form of the offspring generating functions of particles that the probability of survival up to generation $n$ of the process initiated at moment zero by a single particle of type $i$ is equivalent to $c_{i}n^{-1/2}$ for large $n,$ where $c_{i}$ is a positive constant. Earlier such asymptotic representation was known only for the case of the fractional linear offspring gener…
We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.
This is a survey on recent progresses in the study of branching processes with immigration, generalized Ornstein-Uhlenbeck processes and affine Markov processes. We mainly focus on the applications of skew convolution semigroups and the connections in those processes.
We observe the continuous-time Markov Branching Process without high-order moments and allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. Main mathematical tool is regularly varying generating functions with remainder.
We construct a continuous state branching process with immigration (CBI) whose immigration depends on the CBI itself and we recover a continuous state branching process (CB). This provides a dual construction of the pruning at nodes of CB introduced by the authors in a previous paper. This construction is a natural way to model neutral mutation. Using exponential formula, we compute the probability of extinction of the original type population in a critical or sub-critical quadratic branching, conditionally on the non extinction of the total population.
In this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well-known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.
Detection of extremely rare variant alleles, such as tumour DNA, within a complex mixture of DNA molecules is experimentally challenging due to sequencing errors. Barcoding of target DNA molecules in library construction for next-generation sequencing provides a way to identify and bioinformatically remove polymerase induced errors. During the barcoding procedure involving $t$ consecutive PCR cycles, the DNA molecules become barcoded by unique molecular identifiers (UMI). Different library construction protocols utilise different values of $t$. The effect of a larger $t$ and imperfect PCR ampl…
Under natural assumptions a Feller type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.
In this paper, we first form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the hitting time of corresponding random walks.
A multi-type continuous state and continuous time branching process with immigration satisfying some moment conditions is identified as a pathwise unique strong solution of certain stochastic differential equation with jumps.
We study the estimation of two-type continuous-state branching processes with immigration (CBI-processes). The ergodicity of the processes is proved. 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.
We consider the subcritical contact branching random walk on Zd in continuous time with the arbitrary number of offspring and with immigration. We prove the existence of the steady state (statistical equilibrium).
We establish recurrence and transience criteria for critical branching processes in random environment with immigration. These results are then applied to discuss recurrence and transience of a recurrent random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1.
For a positive continuous function f satisfying some standard conditions, we study the f-moments of continuous-state branching processes with or without immigration. The main results give criteria for the existence of the f-moments. The characterization of the processes in terms of stochastic equations given by Dawson and Li (2012) plays an essential role in the proofs.
We study the asymptotic behaviour of the survival probability of a multi-type branching processes in random environment. The class of processes we consider corresponds, in the one-dimensional situation, to the intermediately subcritical case. We show under rather general assumptions on the form of the offspring generating functions of particles that the probability of survival up to generation $n$ of the process initiated at moment zero by a single particle of any type is of order $λ^{n}n^{-1/2}$ for large $n,$ where $λ\in (0,1)$ is a constant specified by the Lyapunov exponent of the mean mat…
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 spe…
Under a fourth order moment condition on the branching and a second order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous state and continuous time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In case of a non-trivial process, under a first order moment condition on the immigration mechanism, we…
In this work we investigate limit theorems for the time-averaged process $\left(\frac{1}{t}\int_0^t X_s^x ds\right)_{t\geq 0}$ where $X^x$ is a subcritical continuous-state branching processes with immigration (CBI processes) starting in $x \geq 0$. Under a second moment condition on the branching and immigration measures we first prove the law of large numbers in $L^2$ and afterward establish the central limit theorem. Assuming additionally that the big jumps of the branching and immigration measures have finite exponential moments of some order, we prove in our main result the large deviatio…
Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.
We consider a supercritical Galton-Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit ${\mathcal{W}}$ for the normalized population size. Small value probabilities for ${\mathcal{W}}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized.
Let (Zn) be a branching process with immigration in an independent and identically distributed random environment. Under necessary moment conditions, we show the exact convergence rate in the central limit theorem on logZn by using the convergence rates of the logarithm of submartingale and the result of the corresponding random walk on the Berry-Esseen bound.
We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter $…
A critical branching process with immigration which evolve in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the aboriginal population we investigate the tail distribution of the so-called life period of the process, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.
In this paper, we consider $n$-type Markov branching processes with immigration and resurrection. The uniqueness criteria are first established. Then, a new method is found and the explicit expression of extinction probability is successfully obtained in the absorption case, the mean extinction time is also given. The recurrence and ergodicity criteria are given if the state ${\bf 0}$ is not absorptive. Finally, if the resurrection rates are same as the immigration rates, the branching property and decay property are discussed in detail, it is shown that the process is a superimposition of a $…
Recursions for moments of multi-type continuous state and continuous time branching process with immigration are derived. It turns out that the $k$-th (mixed) moments and the $k$-th (mixed) central moments are polynomials of the initial value of the process, and their degree are at most $k$ and $\lfloor k/2 \rfloor$, respectively.
