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 process with immigration. These assumptions underlying our weak convergence arguments are easily seen to be met in the Brownian setting, and more generally an $α$-stable setting for any $α\in(1,2]$.
We show the existance of the stationary distributions of subcritical multitype Galton-Watson processes without any conditions on the mean matrix of the offspring distributions. Some additional properties of the stationary distribution are also proven.
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.
We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton--Watson trees with a countable number of types, thus extending a result of G. Miermont on multitype Galton--Watson trees with finitely many types.
In this work, we study asymptotics of multitype Galton-Watson trees with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Lévy process. This gives an analogue of a result obtained by Miermont in the case of multitype Galton-Watson trees with finite covariance matrices of the offsprin…
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 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.
Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated $Q$-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the $Q$-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained $Q$-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype…
Let $A$ be a primitive matrix and let $λ$ be its Perron-Frobenius eigenvalue. We give formulas expressing the associated normalized Perron-Frobenius eigenvector as a simple functional of a multitype Galton-Watson process whose mean matrix is $A$, as well as of a multitype branching process with mean matrix $e^{(A-I)t}$. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.
In this article, we prove a joint large deviation principle in $n$ for the \emph{empirical pair measure} and \emph{ empirical offspring measure} of critical multitype Galton-Watson trees conditioned to have exactly $n$ vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by \cite[Theorem~2.1]{DMS03}. For the case where the offspring law of the tree is a geometric distribution with parameter \sfrac{1}{2}, we get an exact rate function. All o…
We study the bijection between binary Galton--Watson trees in continuous time and their exploration process, both in the sub- and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to infinity. We thus deduce Delmas' generalization of the second Ray--Knight theorem.
We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton-Watson with immigration processes. We also significantly improve some results on multitype size-dependent Galton-Watson processes.
We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős-Rényi edge-addition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a det…
For a discrete time multitype supercritical Galton-Watson process $(Z_n)_{n\in \mathbb{N}}$ and corresponding genealogical tree $\mathbb{T}$, we associate a new discrete time process $(Z_n^Φ)_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^Φ$ is determined by a (random) characteristic $Φ$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^Φ$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $Φ_u$, where $(Φ_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $Φ$. Such processes are known in …
We present two models of multitype Galton-Watson trees, that we call full binary trees and full binary trees with survivals. We show relevant relations between these trees and the Narayana numbers and the two-dimensional decompositions of integers. We prove further statistical results on our models concerning the related contour processes and offspring distributions.
We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component.
We study a classical multitype Galton--Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to $\infty$ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on $k$ digits from the master sequence approaches $$(σe^{-a}-1)\frac{a^k}{k!}\sum_{i\geq1}\frac{i^k}{σ^i}\,,$$ where $σ$ is the selective advantage of the master sequence and $a$ is the product of the …
We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton-Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at st…
We introduce a random finite rooted tree $\mathcal{C}$, the steady state cluster, characterized by a recursive description: $\mathcal{C}$ is a singleton with probability $1/2$ and otherwise is obtained by joining by an edge the roots of two independent trees $\mathcal{C}'$ and $\mathcal{C}''$, each having the law of $\mathcal{C}$, then re-rooting the resulting tree at a uniform random vertex. We construct a stationary regenerative stochastic process $\mathcal{C}(t)$, the steady state cluster growth process. It is characterized by a simple fixed-point property. Its stationary distribution is th…
