Processes In Random Environment Research Articles

Closed-form solutions for Bernoulli and compound Poisson branching processes in random environments*

Abstract For branching processes, the generating functions for the limit distributions of the so-called ratios of probabilities of rare events satisfy Schröder-type integral–functional equations. With the exception of a few special cases, the corresponding equations cannot be solved analytically. I found a large class of Poisson-type offspring distributions for which the Schröder-type functional equations can be solved analytically. Moreover, for the asymptotics of limit distributions, the power and constant factors can be written explicitly. As a bonus, Bernoulli branching processes in random environments are treated. The beauty of this example is that the explicit formula for the generating function is unknown. Nevertheless, the closed-form expressions for the power and constant factors in the asymptotic can be written with the help of some effective techniques. The asymptotic expansion also contains oscillatory terms absent in the Poisson case. It is shown that at power three in the Bernoulli binomial kernels, short-phase discrete oscillations turn into long-phase continuous ones.

From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment

Consider (Zn)n⩾0 a supercritical branching process in an independent and identically distributed environment. Based on some recent development in martingale limit theory, we established law of the iterated logarithm, strong law of large numbers, invariance principle and optimal convergence rate in the central limit theorem under Zolotarev and Wasserstein distances of order p∈(0,2] for the process (logZn)n⩾0.

Quenched large deviations in renewal theory

In this paper we introduce and study renewal–reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

Uniform Cramér moderate deviations and Berry–Esseen bounds for superadditive bisexual branching processes in random environments

Uniform Cramér moderate deviations and Berry–Esseen bounds for superadditive bisexual branching processes in random environments

Asymptotic behaviour of the survival probability of almost critical branching processes in a random environment

A generalization of the well-known result concerning the survival probability of a critical branching process in random environment $Z_k$ is considered. The triangular array scheme of branching processes in random environment $Z_{k,n}$ that are close to $Z_k$ for large $n$ is studied. The equivalence of the survival probabilities for the processes $Z_{n,n}$ and $Z_n$ is obtained under rather natural assumptions on the closeness of $Z_{k,n}$ and $Z_k$. Bibliography: 7 titles.

Probability Inequalities for Weighted Branching Processes in Random Environments

Probability Inequalities for Weighted Branching Processes in Random Environments

Branching processes in random environments with thresholds

AbstractMotivated by applications to COVID dynamics, we describe a model of a branching process in a random environment $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(\tau_j, \nu_j)\}$ —specifically the values of the process at crossing times, viz. $\{(Z_{\tau_j}, Z_{\nu_j})\}$ —along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distributions of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.

Birth Death Swap population in random environment and aggregation with two timescales

This paper deals with the stochastic modeling of a class of heterogeneous population in random environment, structured by discrete subgroups, and called birth–death–swap. In addition to demographic events modifying the population size, swap events, i.e. moves between subgroups, occur in the population. Event intensities are random functionals of the population. In the first part, we show that the complexity of the problem is significantly reduced by modeling the jump measure of the population as a multivariate counting process. This process is defined as the solution of a stochastic differential system with random coefficients, driven by a multivariate Poisson random measure. The solution is obtained under weak assumptions, by the thinning of a strongly dominating point process generated by the same Poisson measure. This key construction relies on a general strong comparison result, of independent interest.The second part is dedicated to averaging results when swap events are significantly more frequent than demographic events. An important ingredient is the stable convergence, which is well-adapted to the general random environment. The pathwise construction by domination yields tightness results straightforwardly. At the limit, the demographic intensity functionals are averaged against random kernels resulting from swap events. Finally, under a natural assumption, we show the convergence of the aggregated population to a birth–death process in random environment, with non-linear intensity functionals.

Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants

Abstract We consider the branching process Z n = X n , 1 + ⋯ + X n Z n − 1 $ Z_{n} =X_{n, 1} + \dotsb +X_{nZ_{n-1}} $ , in random environments η , where η is a sequence of independent identically distributedvariables, for fixed η the random variables X i, j areindependent, have the geometric distribution. We suppose that the associated random walk S n = ξ 1 + ⋯ + ξ n $ S_n = \xi_1 + \dotsb + \xi_n $ has positive meanμ,0 < h<h +satisfies the right-hand Cramer’s condition E exp(h ξ i ) < ∞ for, some h +. Under theseassumptions, we find the asymptotic representation for local probabilities P (Z n =⌊exp(θ n)⌋) for θ ∈ [θ 1, θ 2]⊂</given−names><x> </x><surname>(μ;μ +) and someμ +.

Scaling limits of bisexual Galton–Watson processes

Bisexual Galton–Watson processes are discrete Markov chains where reproduction events are due to mating of males and females. Owing to this interaction, the standard branching property of Galton–Watson processes is lost. We prove tightness for conveniently rescaled bisexual Galton–Watson processes, based on recent techniques developed in [V. Bansaye, M.E. Caballero, and S. Méléard, Scaling limits of population and evolution processes in random environment, Electron. J. Probab. 24(19) (2019), pp. 1–38]. We also identify the possible limits of these rescaled processes as solutions of a stochastic system, coupling two equations through singular coefficients in Poisson terms added to square roots as coefficients of Brownian motions. Under some additional integrability assumptions, pathwise uniqueness of this limiting system of stochastic differential equations and convergence of the rescaled processes are obtained. Two examples corresponding to mutual fidelity are considered.

Critical branching processes in a sparse random environment

We introduce a branching process in a sparse random environment as an intermediate model between a Galton–Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.

Branching processes in random environment with cooling

Известно, что ветвящийся процесс в случайной среде тесно связан с соответствующим случайным блужданием $S_n = \xi_1 + \dotsb + \xi_n$, где $\xi_k = \ln \varphi_{\eta_k}'(1)$. Здесь $\varphi_x (t)$ и $\{ \eta_k \}_{k = 1}^{\infty}$ - производящая функция числа потомков и случайная среда соответственно. В статье изучается вероятность вырождения ветвящегося процесса в случайной среде с замораживаниями при $\mathsf{E} \xi_1 > 0$, отличающегося от обычного ветвящегося процесса в случайной среде тем, что каждое значение среды устанавливается на несколько поколений. Оказывается, что такой процесс также тесно связан со случайным блужданием $S_n = \tau_1 \xi_1 + \dotsb + \tau_n \xi_n$, где $\xi_k = \ln \varphi_{\eta_k}'(1)$. Здесь $\varphi_x (t)$ и $\{ \eta_k \}_{k = 1}^{\infty}$ - производящая функция числа потомков одной частицы при условии среды $x$ и случайная среда соответственно, а $\tau_k$ - длительность $k$-го замораживания. Статья содержит несколько условий, достаточных для невырождения процесса с положительной вероятностью, и несколько условий, достаточных для вырождения процесса с вероятностью $1$.

Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants

Abstract We consider local probabilities of lower deviations for branching process Z n = X n , 1 + ⋯ + X n , Z n − 1 ${{Z}_{n}}={{X}_{n,1}}+\cdots +{{X}_{n,{{Z}_{n-1}}}}$ in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables X i,j are geometric ones.We suppose that the associated random walk S n = ξ 1 + ⋯ + ξ n ${{S}_{n}}={{\xi }_{1}}+\cdots +{{\xi }_{n}}$ has positive mean μ and satisfies left-hand Cramer’s condition E exp ( h ξ i ) < ∞ if h − < h < 0 $\mathbf{E}\exp \left( h{{\xi }_{i}} \right)<\infty \text{ if }{{h}^{-}}<h<0$ for some h − < − 1. ${{h}^{-}}<-1.$ Under these assumptions, we find the asymptotic representation of local probabilities P ( Z n = ⌊ exp ( θ n ) ⌋ ) for θ ∈ [ θ 1 , θ 2 ] ⊂ ( μ − ; μ ) $\mathbf{P}\left( {{Z}_{n}}=\left\lfloor \exp (\theta n) \right\rfloor \right)\text{ for }\theta \in \left[ {{\theta }_{1}},{{\theta }_{2}} \right]\subset \left( {{\mu }^{-}};\mu \right)$ for some non-negative μ −.

Limit theorems for supercritical branching processes in random environment

The branching process in random environment {Zn}n≥0 is a population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with positive probability and grows exponentially fast on the nonextinction set. Using Fourier techniques we improve existing central limit theorem as well as we obtain Edgeworth expansions and the renewal theorem for the sequence {logZn}n≥0. The strategy is to compare logZn with partial sums of i.i.d. random variables in order to obtain precise estimates.

Precise large deviation estimates for branching process in random environment

Nous considérons un processus de branchement dans un environnement aléatoire {Zn}n≥0, qui est le processus de croissance d’une population où les individus se reproduisent indépendamment les uns des autres avec la loi de reproduction choisie au hasard à chaque génération. Nous décrivons des asymptotiques précises des grandes déviations supérieures, c’est-à-dire de P[Zn>eρn]. De plus dans le cas sous-critique, sous la condition de Cramér pour la moyenne de la loi de reproduction, nous étudions les estimations des grandes déviations pour le premier temps de passage de ce processus de branchement et de la taille totale de sa population.

Conditional $$L^1$$-Convergence for the Martingale of a Critical Branching Process in Random Environment

Conditional $$L^1$$-Convergence for the Martingale of a Critical Branching Process in Random Environment

Multitype branching processes in random environment

A survey of results in the theory of multitype branching processes evolving in a random environment is presented. Bibliography: 104 titles.

Limit Theorems for a Strongly Supercritical Branching Process with Immigration in Random Environment

Abstract We consider a strongly supercritical branching process in random environment with immigration stopped at a distant time 𝑛. The offspring reproduction law in each generation is assumed to be geometric. The process is considered under the condition of its extinction after time 𝑛. Two limit theorems for this process are proved: the first one is for the time interval from 0 till 𝑛, and the second one is for the time interval from 𝑛 till + ∞ +\infty .

Central limit theorem for a critical multitype branching process in random environments

Let (Z n) n$\ge$0 with Z n = (Z n (i, j)) 1$\le$i,j$\le$p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn $\sqrt$ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.

Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities

Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.