Small value probabilities for supercritical branching processes with immigration

Consider a supercritical d-type branching process $Z_n^{i} = $ $ (Z_n^i(1), \cdots, Z_n^i(d)), \,n\geq 0,$ in an i.i.d. environment $\xi =(\xi_0, \xi_1, \ldots)$, starting with one particle of type $i$, whose offspring distributions of generation $n$ depend on the environment $\xi_n$ at time $n$. In the deterministic environment case, the famous Kesten-Stigum (1966) theorem states essentially that, if the mean matrix of the offspring distribution has spectral radius $\rho >1$, then for all $i,j =1, \ldots, d$, almost surely $ \lim_{n\rightarrow \infty} \frac{Z_{n}^{i}(j)}{\rho^n }$ exists and is finite; moreover, the limit variables are non-degenerate if and only if $\mathbb{E} \left(Z_1^{i}(j) \log^{+} Z_1^{i}(j)\right)<+\infty$ for all $i$ and $j$. The extension to the random environment case with $d=1$ has been done by Athreya and Karlin (1971) and Tanny (1988). Extending the Kesten-Stigum theorem to the random environment case with $d>1$ is a long-standing problem. The main objective of this paper is to resolve this delicate problem. In particular, under simple conditions, we prove that for any $1\le i,j\le d$, as $n \rightarrow +\infty$, $Z_n^{i}(j)/\mathbb{E}_\xi Z_n^{i}(j) \rightarrow W^{i}$ in probability, where $W^i$ is a non-negative random variable, $ \mathbb{E}_\xi Z_n^{i}(j)$ is the conditional expectation of $Z_n^{i}(j)$ given the environment $\xi$, which diverges to $\infty$ with geometric rate in the sense that $ \frac{1}{n} \log \mathbb{E}_\xi Z_n^{i}(j) \rightarrow \gamma >0$ almost surely, $\gamma$ being the Lyapunov exponent of the mean matrices of the offspring distributions; moreover $W^i$ are non-degenerate for all $i $ if and only if $\mathbb{E}\left(\frac{Z_1^{i}(j)}{M_0(i,j)}\log^{+}\frac{Z_1^{i}(j)}{M_0(i,j)}\right)<+\infty$ for all $i$ and $j$, where $M_0(i,j)$ is the conditioned mean of the number of children of type $j$ produced by a particle of type $i$ at time $0$, given the environment $\xi$. The key idea of the proof is the introduction of a non-negative martingale $(W_n^{i})$ which converges a.s. to $W^i$, and which reduces to the well-known fundamental martingale in the deterministic environment case. In addition, we prove that the direction $Z_n^{i}/ \| Z_n^{i} \|$ converges in law conditioned on the explosion event $\{\|Z_n^i\| \rightarrow +\infty\}$. The case of stationary and ergodic environment is also considered. Our results open ways to prove important properties such as central limit theorems with convergence rate and large deviation asymptotics.

Controlled Branching Processes

Large deviation rates for supercritical and critical branching processes.- How fast does a general branching random walk spread?.- Boltzmann-Gibbs weights in the branching random walk.- Stochastic monotonicity and branching processes.- Multilevel multitype branching models of an information system.- On the shape of the wavefront of branching random walk.- Limiting distributions in branching processes with two types of particles.- Depth-first search of random trees, and Poisson point processes.- Towards dependence in general branching processes.- A criterion of boundedness of discrete branching random walk.- Quasistationarity in a branching model of division-within-division.- Population and density dependent branching processes.- Directed polymers in random media and spin glass models on trees..- A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes.- On two measures defined on the boundary of a branching tree.- Which critically branching populations persist?.- A simple path to Biggins' martingale convergence for branching random walk.- Unsolved problems concerning random walks on trees.- Branching processes with local dependencies.- Sharpness of second moment criteria for branching and tree-indexed processes.- On the recognition and structure of probability generating functions.- Record values of a family of branching processes.- Limit skeleton for critical crump-mode-Jagers branching processes.- Markov cascades.- Limit theorems for branching processes with random migration stopped at zero.

Let Z 0 = l, Z 1,Z 2, … denote a super-critical Galton-Watson branching process with 1 0,\left({m^2 - m} \right)^{\frac{1}{2}} \sigma ^{ - 1} Z_n^{ - \frac{1}{2}} m^m \left({W - W_n } \right)\) is asymptotical normal N(0,1) as n →∞ (Heyde [4]). This result is called a central limit analogue by virtue of the analogy with the convergence of \(\sigma ^{ - 1} n^{\frac{1}{2}} \left({\mu - n^{ - 1} S_n } \right)\) to N(0,1) for a sum S n of independent and identically distributed random variables with mean μ and variance σ2.

The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

Branching processes in a random environment are natural generalisations of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in an iid random environment, where the offspring expectation converges from above to 1. We prove that Haldane’s asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical/subcritical regime. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.

Let \(\mathbf {Z}_{n}=(Z_{n}^{(1)},Z_{n}^{(2)},\cdots ,Z_{n}^{(d)})\) be a d-type (d < ∞) Galton-Watson branching process. For a positive integer k ≥ 2. Pick k individuals at random from the nth generation by simple random sampling without replacement. Trace their lines of descent backward in time till they meet. Let Xn,k be the generation number of the coalescence time of these k individuals of the nth generation. We call the common ancestor of these chosen individuals in the Xn,kth generation their last common ancestor. In this paper, the limit behaviors of the distributions of Xn,k, for any integer k ≥ 2, is studied for the supercritical cases. Also, we investigate the limit distribution of joint distribution of the generation number and the type of the last common ancestor of these randomly chosen individuals and their types in the supercritical case.

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.

This paper concerns the asymptotic behaviour of normalized averaging processes associated with a supercritical, indecomposable Markov branching process. Results wellknown in case of a finite set of types are extended to processes with an arbitrary set of types. The process parameter is allowed to be discrete or continuous.Convergence in the quadratic mean is proved on the basis of a weak form of positive regularity. In this setting, limit variables corresponding to different averaging functions are proportional almost everywhere. The rate of convergence is such that process skeletons, defined by uniform partitions of the parameter set, converge with probability 1. Starting from the almost sure convergence of skeletons, we obtain almost sure convergence of the processes themselves. The final sections deals with properties of the limiting distribution functions, in particular with the possible jump at the origin and the existence of a continuous density everywhere else.Other investigations of supercritical Markov branching processes with an infinite or arbitrary set of types are to be found in [2, 3, 4, 6, 7, 13, 15, 19, 21, 23], and [24].

This paper concerns the asymptotic behaviour of normalized averaging processes associated with a supercritical, indecomposable Markov branching process. Results wellknown in case of a finite set of types are extended to processes with an arbitrary set of types. The process parameter is allowed to be discrete or continuous. Convergence in the quadratic mean is proved on the basis of a weak form of positive regularity. In this setting, limit variables corresponding to different averaging functions are proportional almost everywhere. The rate of convergence is such that process skeletons, defined by uniform partitions of the parameter set, converge with probability 1. Starting from the almost sure convergence of skeletons, we obtain almost sure convergence of the processes themselves. The final sections deals with properties of the limiting distribution functions, in particular with the possible jump at the origin and the existence of a continuous density everywhere else. Other investigations of supercritical Markov branching processes with an infinite or arbitrary set of types are to be found in [2, 3, 4, 6, 7, 13, 15, 19, 21, 23], and [24].

Let (Zn) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size Wn converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Zn.

We consider the model of alternating branching processes, where two Markov branching processes act alternately by the random time periods of observation and treatment. Stationary distributions can be obtained by feed-back control if the observation time δ is defined by the additive functional of total progeny of the supercritical Markov branching process ξ(t), or by the explicit immigration of particles. We investigate the reproduction by n cycles and limit theorems to obtain the stationary distributions.

Supercritical Branching Processes with Countably Many Types and the Size of Random Cantor Sets

A branching process with immigration of the following type is considered. For everyi, a random numberNiof particles join the system at time. These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f.and life time distributionG(t). Assume. Then it is shown thatZ(t)e–αtconverges in distribution to an extended real-valued random variableYwhereais the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

Upper deviation results are obtained for the split time of a supercritical continuous-time Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times, which shows that the scaling is completely different from the upper deviations.