Simulations and a conditional limit theorem for intermediately subcritical branching processes in random environment

Intermediately subcritical branching processes in random environment are at the borderline between two subcritical regimes and exhibit particularly rich behavior. In this paper, we prove a functional limit theorem for these processes. It is discussed together with two other recently proved limit theorems for the intermediately subcritical case and illustrated by several computer simulations.     

Surviving particles for subcritical branching processes in random environment

The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives a.s$a.s$. These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE.     

High level subcritical branching processes in a random environment

A subcritical branching process in a random environment is considered under the assumption that the moment-generating function of a step of the associated random walk Θ(t), t ≥ 0, is equal to 1 for some value of the argument ? > 0. Let T _x be the time when the process first attains the half-axis (x,+∞) and T be the lifetime of this process. It is shown that the random variable T _x /lnx, considered under the condition T _x < +∞, converges in distribution to a degenerate random variable equal to 1/Θ′(?), and the random variable T/ ln x, considered under the same condition, converges in distribution to a degenerate random variable equal to 1/Θ′(?) ? 1/Θ′(0).     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

Central limit theorems for a supercritical branching process in a random environment

For a supercritical branching process (Z_n) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population W_n=Z_n/E[Z_n|ξ] to its limit W_∞: we show a central limit theorem for W_∞−W_n with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for W_n+k−W_n for each fixed k∈N^∗.     

Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions

In the present paper, we characterize the behavior of supercritical branching processes in random environment with linear fractional offspring distributions, conditioned on having small, but positive values at some large generation. As it has been noticed in previous works, there is a phase transition in the behavior of the process. Here, we examine the strongly and intermediately supercritical regimes The main result is a conditional limit theorem for the rescaled associated random walk in the intermediately case.     

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution {p_n} assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp_nnlog⁺log⁺n< ∞, then almost sure convergence obtains in the supercritical case.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

Central limit theorems for supercritical branching Markov processes

In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatial-dependent branching mechanisms. These are generalizations of the spatial central limit theorems proved in [1] for branching OU processes with binary branching mechanisms. Compared with the results of [1], our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are non-degenerate.     

Almost critical branching processes and limit theorems

We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 127–133, January, 2009.     

Functional limit theorems for random regular graphs

Consider $d$ uniformly random permutation matrices on $n$ labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree $2d$ on $n$ vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as $n$ grows to infinity, either when $d$ is kept fixed or grows slowly with $n$ . In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein’s method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn–Szemerédi argument for estimating the second largest eigenvalue for all values of $d$ and $n$ .     

Second order limit theorems for the Markov branching process in random environments

In a Markov branching process with random environments, limiting fluctuations of the population size arise from the changing environment, which causes random variation of the ‘deterministic’ population prediction, and from the stochastic wobble around this ‘deterministic’ mean, which is apparent in the ordinary Markov branching process. If the random environment is generated by a suitable stationary process, the first variation typically swamps the second kind. In this paper, environmental processes are considered which, in contrast, lead to sampling and environmental fluctuation of comparable magnitude. The method makes little use either of stationarity or of the branching property, and is amenable to some generalization away from the Markov branching process.     

Transition phenomena for branching processes in a random environment

The behavior of Galton-Watson processes in a random environment in the case of state-dependent immigration and also in the case of state-dependent migration is studied. Limit theorems are obtained in the near-critical case. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

Functional limit theorems for cumulative processes and stopping times

Summary Consider a cumulative regenerative process with increments between regeneration points being i.i.d. r.v.'s. Let the d.f. of those increments belong to the domain of attraction of a stable distribution with exponent less than two. A functional limit theorem in the Skorohod M ₁-topology is proved for this process. The M ₁-topology is more useful than the J ₁-topology in this case, because it allows the cumulative process to be continuous.The second part of the paper concerns a stopping time process, (t)--inf(s>0:w(s)>tg(s)), where w(t) is a process with positive drift for which a functional limit theorem holds and g(t)=t ^p L(t) with 0p<1 and L(t) varying slowly at infinity. Weak convergence for the process (t) is proved under certain conditions in the J ₁- and M ₁-topologies.     

Central limit theorems for stochastic processes under random entropy conditions

Summary Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array ( _nj ) of stochastic processes, indexed by a set S, to a sample-continuous Gaussian process, when the array satisfies a random entropy condition, analogous to one used by Giné and Zinn (1984) for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-ervonenkis property and each _nj (f)fdnj is of the form njc for some reasonable random finite signed measure v _nj. As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik-ervonenkis classes. Special cases include Prokhorov's (1956) central limit theorem for empirical processes, and Shorack's (1979) theorems on weighted empirical processes.Research supported by an NSF Postdoctoral Fellowship, grant no. MCS83-111686