Supercritical Branching Process Research Articles

On the regularity of the extinction probability of a branching process in varying and random environments

We consider a supercritical branching process in time-dependent environment ξ. We assume that the offspring distributions depend regularly (Ck or real-analytically) on real parameters λ. We show that the extinction probability qλ(ξ), given the environment ξ ‘inherits’ this regularity whenever the offspring distributions satisfy a condition of contraction-type. Our proof makes use of the Poincaré metric on the complex unit disc and a real-analytic implicit function theorem.

Genealogy for supercritical branching processes

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

Genealogy for supercritical branching processes

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

Corrections and acknowledgment for “Local limit theory and large deviations for supercritical branching processes”

Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes'' [math.PR/0407059]

THE ASYMPTOTIC PROPERTIES OF SUPERCRITICAL BISEXUAL GALTON-WATSON BRANCHING PROCESSES WITH IMMIGRATION OF MATING UNITS

In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L 1 convergence are given for the process with the suitably normed condition.

Phase Transitions in Dynamical Random Graphs

We study a large-time limit of a Markov process whose states are finite graphs. The number of the vertices is described by a supercritical branching process, and the dynamics of edges is determined by the rates of appending and deleting. We find a phase transition in our model similar to the one in the random graph model G n,p . We derive a formula for the line of critical parameters which separates two different phases: one is where the size of the largest component is proportional to the size of the entire graph, and another one, where the size of the largest component is at most logarithmic with respect to the size of the entire graph. In the supercritical phase we find the asymptotics for the size of the largest component.

On the transience of processes defined on Galton–Watson trees

We introduce a simple technique for proving the transience of certain processes defined on the random tree $\mathcal{G}$ generated by a supercritical branching process. We prove the transience for once-reinforced random walks on $\mathcal{G}$, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567–592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on $\mathcal{G}$. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229–1241] and [Ann. Probab. 18 (1990) 931–958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42–62] proved that a vertex-reinforced jump process defined on the b-ary tree is transient if b≥4 and recurrent if b=1. The case b=2 is still open.

Large scale localization of a spatial version of Neveu’s branching process

Recently a spatial version of Neveu’s (1992) continuous-state branching process was constructed by Fleischmann and Sturm (2004). This superprocess with infinite mean branching behaves quite differently from usual supercritical spatial branching processes. In fact, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric stable motion process.

On the Mean Distance in Scale Free Graphs

We consider a graph, where the nodes have a pre-described degree distribution F, and where nodes are randomly connected in accordance to their degree. Based on a recent result (R. van der Hofstad, G. Hooghiemstra and P. Van Mieghem, “Random graphs with finite variance degrees,” Random Structures and Algorithms, vol. 17(5) pp. 76–105, 2005), we improve the approximation of the mean distance between two randomly chosen nodes given by M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distribution and their application,” Physical Review. E vol. 64, 026118, pp. 1–17, 2001. Our new expression for the mean distance involves the expectation of the logarithm of the limit of a super-critical branching process. We compare simulations of the mean distance with the results of Newman et al. and with our new approach.

Dimensions of supercritical branching processes in varying environments

Let ∂ Γ be the boundary of a family tree Γ associated with a supercritical branching process in varying environments. In this paper, the Hausdorff dimension, the upper box dimension and the packing dimension of ∂ Γ are computed explicitly. In contrast to the (fixed environment) Galton–Watson case, the Hausdorff and upper box dimension may take different values.

Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process

We consider a single-type supercritical or near-critical size-dependent branching process { N n } n such that the offspring mean converges to a limit m ⩾ 1 with a rate of convergence of order N n α as the population size N n grows to ∞ and the variance may change at the rate N n β , where α > 0 and − 1 ⩽ β < 1 . The offspring mean depends on an unknown parameter θ 0 that we estimate on the non-extinction set by using the conditional least squares method. We prove the strong consistency of the estimator of θ 0 as n → ∞ under some general conditions on the asymptotic behavior of the process. We also give its asymptotic distribution for a certain class of size-dependent branching processes. To cite this article: N. Lalam, C. Jacob, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Large deviations for supercritical multitype branching processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x α F *[i](x) + o(x α ) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x −β/(1−β) G*[i](x) + o (x −β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] &amp;gt; x) = x δ/(δ−1) H *[i](x) + o(x δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F *, G * and H * are multiplicatively periodic functions.

Large deviations for supercritical multitype branching processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + xαF*[i](x) + o(xα) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x−β/(1−β) G*[i](x) + o (x−β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] &gt; x) = xδ/(δ−1)H*[i](x) + o(xδ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F*, G* and H* are multiplicatively periodic functions.

Local limit theory and large deviations for supercritical Branching processes

In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn:n≥1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P(Zn=vn) as vn↗∞, and use this to study conditional large deviations of {YZn:n≥1}, where Yn satisfies an LDP, particularly of {Zn−1Zn+1:n≥1} conditioned on Zn≥vn.

Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process

We consider a single-type supercritical or near-critical size-dependent branching process {N n } n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size N n grows to ∞ and the variance may vary at the rate where −1 ≤ β &amp;lt; 1. The offspring mean m(N) = m + μN -α + o(N -α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {N h ,…,N n } by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model m θ,ν̂ n (·), where ν̂ n is any estimation of the parameter of the nuisance part (O(N -α) if θ0 = m and o(N -α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process

We consider a single-type supercritical or near-critical size-dependent branching process {Nn}n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size Nn grows to ∞ and the variance may vary at the rate where −1 ≤ β &lt; 1. The offspring mean m(N) = m + μN-α + o(N-α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {Nh,…,Nn} by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model mθ,ν̂n(·), where ν̂n is any estimation of the parameter of the nuisance part (O(N-α) if θ0 = m and o(N-α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

Supercritical multitype branching processes: the ancestral types of typical individuals

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

Supercritical multitype branching processes: the ancestral types of typical individuals

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

Supercritical branching processes and the role of fluctuations under exponential population growth

We study some exact properties of supercritical branching processes. A proper rescaling of the relevant variable allows us to determine the distribution of population sizes after a number of generations have elapsed. Both time-continuous and discrete processes are analysed and compared. The obtained results are of relevance for the growth of populations that are not resource limited (a typical situation in some biological processes that can be modelled by laboratory experiments). Large fluctuations inherent to the process play a main role when bottlenecks occur.

Harmonic moments and large deviation rates for supercritical branching processes

Let $ \{Z_{n}, n \ge 1 \}$ be a single type supercritical Galton--Watson process with mean $EZ_{1} \equiv m$, initiated by a single ancestor. This paper studies the large deviation behavior of the sequence $\{R_n \equiv \frac{Z_{n+1}}{Z_n}\dvtx n \ge 1 \}$ and establishes a "phase transition" in rates depending on whether $r$, the maximal number of moments possessed by the offspring distribution, is less than, equal to or greater than the Schröder constant $\alpha$. This is done via a careful analysis of the harmonic moments of $Z_n$.